Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.4%

Time: 10.3s

Alternatives: 20

Speedup: 1.4×

• 49.8% 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: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (* K 0.5))))
   (if (<= t_0 (- INFINITY))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (if (<= t_0 2e-15)
       (+ U (* l (* J (* 2.0 t_1))))
       (+ U (* t_0 (* J t_1)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K * 0.5));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else if (t_0 <= 2e-15) {
		tmp = U + (l * (J * (2.0 * t_1)));
	} else {
		tmp = U + (t_0 * (J * t_1));
	}
	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 t_1 = Math.cos((K * 0.5));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else if (t_0 <= 2e-15) {
		tmp = U + (l * (J * (2.0 * t_1)));
	} else {
		tmp = U + (t_0 * (J * t_1));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	elif t_0 <= 2e-15:
		tmp = U + (l * (J * (2.0 * t_1)))
	else:
		tmp = U + (t_0 * (J * t_1))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	elseif (t_0 <= 2e-15)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * t_1))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	elseif (t_0 <= 2e-15)
		tmp = U + (l * (J * (2.0 * t_1)));
	else
		tmp = U + (t_0 * (J * t_1));
	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]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 2e-15], N[(U + N[(l * N[(J * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   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))) < 2.0000000000000002e-15

   1. Initial program 68.6%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 2.0000000000000002e-15 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-15)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* l (* J (* 2.0 (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-15)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * (J * (2.0 * 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-15)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * (J * (2.0 * 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-15):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (l * (J * (2.0 * 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-15))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * 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-15)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (l * (J * (2.0 * 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-15]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(J * N[(2.0 * 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 2.0000000000000002e-15 < (-.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))) < 2.0000000000000002e-15

   1. Initial program 68.6%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.955:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot t\_0\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.92:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot {K}^{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.955)
     (+ U (* l (* J (* 2.0 t_0))))
     (if (<= t_1 -0.92)
       (+ U (* (* l J) (+ 2.0 (* -0.25 (pow K 2.0)))))
       (if (<= t_1 0.15)
         (+ U (* 2.0 (* J (* l t_0))))
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.955) {
		tmp = U + (l * (J * (2.0 * t_0)));
	} else if (t_1 <= -0.92) {
		tmp = U + ((l * J) * (2.0 + (-0.25 * pow(K, 2.0))));
	} else if (t_1 <= 0.15) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = cos((k / 2.0d0))
    if (t_1 <= (-0.955d0)) then
        tmp = u + (l * (j * (2.0d0 * t_0)))
    else if (t_1 <= (-0.92d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((-0.25d0) * (k ** 2.0d0))))
    else if (t_1 <= 0.15d0) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.955) {
		tmp = U + (l * (J * (2.0 * t_0)));
	} else if (t_1 <= -0.92) {
		tmp = U + ((l * J) * (2.0 + (-0.25 * Math.pow(K, 2.0))));
	} else if (t_1 <= 0.15) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -0.955:
		tmp = U + (l * (J * (2.0 * t_0)))
	elif t_1 <= -0.92:
		tmp = U + ((l * J) * (2.0 + (-0.25 * math.pow(K, 2.0))))
	elif t_1 <= 0.15:
		tmp = U + (2.0 * (J * (l * t_0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.955)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * t_0))));
	elseif (t_1 <= -0.92)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(-0.25 * (K ^ 2.0)))));
	elseif (t_1 <= 0.15)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_1 <= -0.955)
		tmp = U + (l * (J * (2.0 * t_0)));
	elseif (t_1 <= -0.92)
		tmp = U + ((l * J) * (2.0 + (-0.25 * (K ^ 2.0))));
	elseif (t_1 <= 0.15)
		tmp = U + (2.0 * (J * (l * t_0)));
	else
		tmp = U + (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[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.955], N[(U + N[(l * N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.92], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(-0.25 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.95499999999999996

   1. Initial program 65.8%

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

   3. Taylor expanded in l around 0 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-*r* 59.8%

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

      3. associate-*l* 59.8%

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

      4. *-commutative 59.8%

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

      5. *-commutative 59.8%

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

      6. associate-*l* 59.8%

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

      7. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   if -0.95499999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.92000000000000004

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 2.5%

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

   7. Taylor expanded in K around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 84.2%

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

      5. associate-*l* 84.2%

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

      6. *-commutative 84.2%

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

      7. associate-*l* 84.2%

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

      8. distribute-lft-out 84.2%

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

   9. Simplified 84.2%

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

   if -0.92000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.149999999999999994

   1. Initial program 91.4%

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

   3. Taylor expanded in l around 0 65.1%

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

   if 0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 83.9%

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

   3. Taylor expanded in l around 0 90.0%

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

   4. Taylor expanded in K around 0 87.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.955:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.92:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot {K}^{2}\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.15:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.135:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.55:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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) U)))
   (if (<= l -1.8e+83)
     t_0
     (if (<= l -0.135)
       t_1
       (if (<= l 0.55)
         (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
         (if (<= l 2.4e+84) 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) + U;
	double tmp;
	if (l <= -1.8e+83) {
		tmp = t_0;
	} else if (l <= -0.135) {
		tmp = t_1;
	} else if (l <= 0.55) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else if (l <= 2.4e+84) {
		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) + u
    if (l <= (-1.8d+83)) then
        tmp = t_0
    else if (l <= (-0.135d0)) then
        tmp = t_1
    else if (l <= 0.55d0) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else if (l <= 2.4d+84) 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) + U;
	double tmp;
	if (l <= -1.8e+83) {
		tmp = t_0;
	} else if (l <= -0.135) {
		tmp = t_1;
	} else if (l <= 0.55) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else if (l <= 2.4e+84) {
		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) + U
	tmp = 0
	if l <= -1.8e+83:
		tmp = t_0
	elif l <= -0.135:
		tmp = t_1
	elif l <= 0.55:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	elif l <= 2.4e+84:
		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(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1.8e+83)
		tmp = t_0;
	elseif (l <= -0.135)
		tmp = t_1;
	elseif (l <= 0.55)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	elseif (l <= 2.4e+84)
		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) + U;
	tmp = 0.0;
	if (l <= -1.8e+83)
		tmp = t_0;
	elseif (l <= -0.135)
		tmp = t_1;
	elseif (l <= 0.55)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	elseif (l <= 2.4e+84)
		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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.8e+83], t$95$0, If[LessEqual[l, -0.135], t$95$1, If[LessEqual[l, 0.55], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4e+84], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.7999999999999999e83 or 2.4e84 < 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 98.9%

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

   4. Taylor expanded in l around inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 98.9%

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

   6. Simplified 98.9%

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

   if -1.7999999999999999e83 < l < -0.13500000000000001 or 0.55000000000000004 < l < 2.4e84

   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 65.8%

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

   if -0.13500000000000001 < l < 0.55000000000000004

   1. Initial program 69.1%

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

   3. Taylor expanded in l around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.4%

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

      3. associate-*l* 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*l* 99.4%

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

      7. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+83}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.135:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.55:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+84}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+84}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.135:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -3.7e+84)
     (+ U (* t_0 (* J (* 0.3333333333333333 (pow l 3.0)))))
     (if (<= l -0.135)
       (+ (* (- (exp l) (exp (- l))) J) U)
       (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= -3.7e+84) {
		tmp = U + (t_0 * (J * (0.3333333333333333 * pow(l, 3.0))));
	} else if (l <= -0.135) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (t_0 * (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) :: tmp
    t_0 = cos((k / 2.0d0))
    if (l <= (-3.7d+84)) then
        tmp = u + (t_0 * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    else if (l <= (-0.135d0)) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (t_0 * (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.cos((K / 2.0));
	double tmp;
	if (l <= -3.7e+84) {
		tmp = U + (t_0 * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	} else if (l <= -0.135) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= -3.7e+84:
		tmp = U + (t_0 * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	elif l <= -0.135:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -3.7e+84)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))));
	elseif (l <= -0.135)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(t_0 * 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 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -3.7e+84)
		tmp = U + (t_0 * (J * (0.3333333333333333 * (l ^ 3.0))));
	elseif (l <= -0.135)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (t_0 * (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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.7e+84], N[(U + N[(t$95$0 * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -0.135], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.7e84

   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 98.1%

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

   4. Taylor expanded in l around inf 98.1%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 98.1%

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

   6. Simplified 98.1%

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

   if -3.7e84 < l < -0.13500000000000001

   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 65.0%

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

   if -0.13500000000000001 < l

   1. Initial program 78.8%

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

   3. Taylor expanded in l around 0 93.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+84}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.135:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.15:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.15)
   (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.15) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.15d0) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.15) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.15:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.15)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.15)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.15], N[(U + N[(l * N[(J * N[(2.0 * 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[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.149999999999999994

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*r* 58.9%

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

      3. associate-*l* 58.9%

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

      4. *-commutative 58.9%

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

      5. *-commutative 58.9%

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

      6. associate-*l* 58.9%

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

      7. *-commutative 58.9%

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

   5. Simplified 58.9%

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

   if 0.149999999999999994 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 83.9%

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

   3. Taylor expanded in l around 0 90.0%

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

   4. Taylor expanded in K around 0 87.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.15:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.14 \lor \neg \left(\ell \leq 0.84\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.14) (not (<= l 0.84)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.14) || !(l <= 0.84)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.14d0)) .or. (.not. (l <= 0.84d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.14) || !(l <= 0.84)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.14) or not (l <= 0.84):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.14) || !(l <= 0.84))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.14) || ~((l <= 0.84)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.14], N[Not[LessEqual[l, 0.84]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.14000000000000001 or 0.839999999999999969 < 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 72.1%

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

   if -0.14000000000000001 < l < 0.839999999999999969

   1. Initial program 69.1%

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

   3. Taylor expanded in l around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.4%

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

      3. associate-*l* 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*l* 99.4%

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

      7. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.14 \lor \neg \left(\ell \leq 0.84\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+41}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+61}:\\ \;\;\;\;U + \left(-0.25 \cdot \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (*
          U
          (+
           1.0
           (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))))
   (if (<= l -1.05e+22)
     t_0
     (if (<= l 2.7e+41)
       (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
       (if (<= l 3.35e+61)
         (+ U (+ (* -0.25 (* J (* l (pow K 2.0)))) (* 2.0 (* l J))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / U));
	double tmp;
	if (l <= -1.05e+22) {
		tmp = t_0;
	} else if (l <= 2.7e+41) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else if (l <= 3.35e+61) {
		tmp = U + ((-0.25 * (J * (l * pow(K, 2.0)))) + (2.0 * (l * J)));
	} 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 * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    if (l <= (-1.05d+22)) then
        tmp = t_0
    else if (l <= 2.7d+41) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else if (l <= 3.35d+61) then
        tmp = u + (((-0.25d0) * (j * (l * (k ** 2.0d0)))) + (2.0d0 * (l * j)))
    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 * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	double tmp;
	if (l <= -1.05e+22) {
		tmp = t_0;
	} else if (l <= 2.7e+41) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else if (l <= 3.35e+61) {
		tmp = U + ((-0.25 * (J * (l * Math.pow(K, 2.0)))) + (2.0 * (l * J)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	tmp = 0
	if l <= -1.05e+22:
		tmp = t_0
	elif l <= 2.7e+41:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	elif l <= 3.35e+61:
		tmp = U + ((-0.25 * (J * (l * math.pow(K, 2.0)))) + (2.0 * (l * J)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)))
	tmp = 0.0
	if (l <= -1.05e+22)
		tmp = t_0;
	elseif (l <= 2.7e+41)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	elseif (l <= 3.35e+61)
		tmp = Float64(U + Float64(Float64(-0.25 * Float64(J * Float64(l * (K ^ 2.0)))) + Float64(2.0 * Float64(l * J))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	tmp = 0.0;
	if (l <= -1.05e+22)
		tmp = t_0;
	elseif (l <= 2.7e+41)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	elseif (l <= 3.35e+61)
		tmp = U + ((-0.25 * (J * (l * (K ^ 2.0)))) + (2.0 * (l * J)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = 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]}, If[LessEqual[l, -1.05e+22], t$95$0, If[LessEqual[l, 2.7e+41], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.35e+61], N[(U + N[(N[(-0.25 * N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.0499999999999999e22 or 3.34999999999999972e61 < 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 85.3%

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

   4. Taylor expanded in K around 0 64.5%

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

   5. Taylor expanded in U around inf 67.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   if -1.0499999999999999e22 < l < 2.7e41

   1. Initial program 71.8%

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

   3. Taylor expanded in l around 0 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*r* 92.6%

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

      3. associate-*l* 92.6%

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

      4. *-commutative 92.6%

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

      5. *-commutative 92.6%

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

      6. associate-*l* 92.6%

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

      7. *-commutative 92.6%

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

   5. Simplified 92.6%

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

   if 2.7e41 < l < 3.34999999999999972e61

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 4.8%

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

   7. Taylor expanded in K around 0 83.4%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+41}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+61}:\\ \;\;\;\;U + \left(-0.25 \cdot \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+21} \lor \neg \left(\ell \leq 13.8\right):\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3e+21) (not (<= l 13.8)))
   (* U (+ 1.0 (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))
   (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3e+21) || !(l <= 13.8)) {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / U));
	} else {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-3d+21)) .or. (.not. (l <= 13.8d0))) then
        tmp = u * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    else
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3e+21) || !(l <= 13.8)) {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	} else {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3e+21) or not (l <= 13.8):
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	else:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3e+21) || !(l <= 13.8))
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)));
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -3e+21) || ~((l <= 13.8)))
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	else
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3e+21], N[Not[LessEqual[l, 13.8]], $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], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3e21 or 13.800000000000001 < 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 80.6%

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

   4. Taylor expanded in K around 0 60.3%

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

   5. Taylor expanded in U around inf 62.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   if -3e21 < l < 13.800000000000001

   1. Initial program 70.3%

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

   3. Taylor expanded in l around 0 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*r* 95.8%

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

      3. associate-*l* 95.8%

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

      4. *-commutative 95.8%

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

      5. *-commutative 95.8%

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

      6. associate-*l* 95.8%

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

      7. *-commutative 95.8%

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

   5. Simplified 95.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+21} \lor \neg \left(\ell \leq 13.8\right):\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+32}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 95:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.8e+32)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 95.0)
     (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
     (* J (* 0.3333333333333333 (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.8e+32) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 95.0) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else {
		tmp = J * (0.3333333333333333 * pow(l, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-5.8d+32)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 95.0d0) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else
        tmp = j * (0.3333333333333333d0 * (l ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.8e+32) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 95.0) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else {
		tmp = J * (0.3333333333333333 * Math.pow(l, 3.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.8e+32:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 95.0:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	else:
		tmp = J * (0.3333333333333333 * math.pow(l, 3.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.8e+32)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 95.0)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.8e+32)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 95.0)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	else
		tmp = J * (0.3333333333333333 * (l ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.8e+32], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 95.0], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.80000000000000006e32

   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 86.0%

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

   4. Taylor expanded in K around 0 61.6%

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

   5. Taylor expanded in l around inf 61.7%

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

   if -5.80000000000000006e32 < l < 95

   1. Initial program 71.4%

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

   3. Taylor expanded in l around 0 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*r* 93.2%

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

      3. associate-*l* 93.2%

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

      4. *-commutative 93.2%

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

      5. *-commutative 93.2%

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

      6. associate-*l* 93.2%

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

      7. *-commutative 93.2%

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

   5. Simplified 93.2%

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

   if 95 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 79.7%

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

   4. Taylor expanded in K around 0 62.0%

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

   5. Taylor expanded in l around inf 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.1%

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

   7. Simplified 62.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+32}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 95:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+32}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 95:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6.8e+32)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 95.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (* J (* 0.3333333333333333 (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.8e+32) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 95.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = J * (0.3333333333333333 * pow(l, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-6.8d+32)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 95.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = j * (0.3333333333333333d0 * (l ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.8e+32) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 95.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = J * (0.3333333333333333 * Math.pow(l, 3.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6.8e+32:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 95.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = J * (0.3333333333333333 * math.pow(l, 3.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6.8e+32)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 95.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -6.8e+32)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 95.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = J * (0.3333333333333333 * (l ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6.8e+32], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 95.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.79999999999999957e32

   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 86.0%

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

   4. Taylor expanded in K around 0 61.6%

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

   5. Taylor expanded in l around inf 61.7%

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

   if -6.79999999999999957e32 < l < 95

   1. Initial program 71.4%

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

   3. Taylor expanded in l around 0 93.2%

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

   if 95 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 79.7%

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

   4. Taylor expanded in K around 0 62.0%

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

   5. Taylor expanded in l around inf 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.1%

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

   7. Simplified 62.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+32}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 95:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+32} \lor \neg \left(\ell \leq 95\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.8e+32) (not (<= l 95.0)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (fma J (* l 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.8e+32) || !(l <= 95.0)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else {
		tmp = fma(J, (l * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.8e+32) || !(l <= 95.0))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	else
		tmp = fma(J, Float64(l * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.8e+32], N[Not[LessEqual[l, 95.0]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.80000000000000006e32 or 95 < 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 61.8%

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

   5. Taylor expanded in l around inf 61.9%

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

   if -5.80000000000000006e32 < l < 95

   1. Initial program 71.4%

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

   3. Taylor expanded in l around 0 93.9%

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

   4. Taylor expanded in K around 0 82.5%

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

   5. Taylor expanded in l around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r* 82.5%

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

      4. fma-define 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+32} \lor \neg \left(\ell \leq 95\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+32}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 95:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.7e+32)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 95.0)
     (fma J (* l 2.0) U)
     (* J (* 0.3333333333333333 (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.7e+32) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 95.0) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = J * (0.3333333333333333 * pow(l, 3.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.7e+32)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 95.0)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.7e+32], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 95.0], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.7e32

   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 86.0%

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

   4. Taylor expanded in K around 0 61.6%

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

   5. Taylor expanded in l around inf 61.7%

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

   if -5.7e32 < l < 95

   1. Initial program 71.4%

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

   3. Taylor expanded in l around 0 93.9%

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

   4. Taylor expanded in K around 0 82.5%

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

   5. Taylor expanded in l around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r* 82.5%

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

      4. fma-define 82.5%

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

   7. Simplified 82.5%

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

   if 95 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 79.7%

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

   4. Taylor expanded in K around 0 62.0%

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

   5. Taylor expanded in l around inf 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.1%

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

   7. Simplified 62.1%

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

Alternative 14: 54.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -850:\\ \;\;\;\;{U}^{-16}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -850.0) (pow U -16.0) (fma J (* l 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -850.0) {
		tmp = pow(U, -16.0);
	} else {
		tmp = fma(J, (l * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -850.0)
		tmp = U ^ -16.0;
	else
		tmp = fma(J, Float64(l * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -850.0], N[Power[U, -16.0], $MachinePrecision], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -850

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{{U}^{-16}} \]

   if -850 < l

   1. Initial program 78.8%

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

   3. Taylor expanded in l around 0 93.3%

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

   4. Taylor expanded in K around 0 80.0%

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

   5. Taylor expanded in l around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. *-commutative 67.8%

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

      3. associate-*r* 67.8%

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

      4. fma-define 67.9%

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

   7. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \ell \cdot 2, U\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 54.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -640:\\ \;\;\;\;{U}^{-16}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -640.0) (pow U -16.0) (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -640.0) {
		tmp = pow(U, -16.0);
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-640.0d0)) then
        tmp = u ** (-16.0d0)
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -640.0) {
		tmp = Math.pow(U, -16.0);
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -640.0:
		tmp = math.pow(U, -16.0)
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -640.0)
		tmp = U ^ -16.0;
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -640.0)
		tmp = U ^ -16.0;
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -640.0], N[Power[U, -16.0], $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -640

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{{U}^{-16}} \]

   if -640 < l

   1. Initial program 78.8%

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

   3. Taylor expanded in J around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. *-commutative 78.8%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in l around 0 77.0%

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

   7. Taylor expanded in K around 0 67.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -640:\\ \;\;\;\;{U}^{-16}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.0% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+173}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -63000000 \lor \neg \left(\ell \leq 1250\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.1e+173)
   (- -4.0 (* U U))
   (if (or (<= l -63000000.0) (not (<= l 1250.0))) (* U U) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.1e+173) {
		tmp = -4.0 - (U * U);
	} else if ((l <= -63000000.0) || !(l <= 1250.0)) {
		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 <= (-1.1d+173)) then
        tmp = (-4.0d0) - (u * u)
    else if ((l <= (-63000000.0d0)) .or. (.not. (l <= 1250.0d0))) 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 <= -1.1e+173) {
		tmp = -4.0 - (U * U);
	} else if ((l <= -63000000.0) || !(l <= 1250.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.1e+173:
		tmp = -4.0 - (U * U)
	elif (l <= -63000000.0) or not (l <= 1250.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.1e+173)
		tmp = Float64(-4.0 - Float64(U * U));
	elseif ((l <= -63000000.0) || !(l <= 1250.0))
		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 <= -1.1e+173)
		tmp = -4.0 - (U * U);
	elseif ((l <= -63000000.0) || ~((l <= 1250.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.1e+173], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -63000000.0], N[Not[LessEqual[l, 1250.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e173

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 20.2%

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

      1. cancel-sign-sub-inv 20.2%

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

   8. Simplified 20.2%

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

   if -1.1e173 < l < -6.3e7 or 1250 < 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 20.1%

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

   if -6.3e7 < l < 1250

   1. Initial program 70.0%

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

   3. Taylor expanded in J around 0 66.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+173}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -63000000 \lor \neg \left(\ell \leq 1250\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.2% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7200000 \lor \neg \left(\ell \leq 960\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7200000.0) (not (<= l 960.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7200000.0) || !(l <= 960.0)) {
		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 <= (-7200000.0d0)) .or. (.not. (l <= 960.0d0))) 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 <= -7200000.0) || !(l <= 960.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7200000.0) or not (l <= 960.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7200000.0) || !(l <= 960.0))
		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 <= -7200000.0) || ~((l <= 960.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7200000.0], N[Not[LessEqual[l, 960.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -7.2e6 or 960 < 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 16.6%

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

   if -7.2e6 < l < 960

   1. Initial program 70.0%

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

   3. Taylor expanded in J around 0 66.4%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7200000 \lor \neg \left(\ell \leq 960\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Taylor expanded in J around 0 84.7%

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

   1. associate-*r* 84.7%

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

   2. *-commutative 84.7%

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

5. Simplified 84.7%

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

6. Taylor expanded in l around 0 61.4%

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

7. Taylor expanded in K around 0 53.5%

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

8. Final simplification 53.5%

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

Alternative 19: 36.0% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Taylor expanded in J around 0 35.1%

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

Alternative 20: 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 84.7%

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

4. Applied egg-rr 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

Reproduce

herbie shell --seed 2024101 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))