Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.8%

Time: 14.8s

Alternatives: 24

Speedup: 1.3×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0.0005\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.0005)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (fma 2.0 l (* 0.3333333333333333 (pow l 3.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.0005)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * fma(2.0, l, (0.3333333333333333 * pow(l, 3.0)))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.0005))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * fma(2.0, l, Float64(0.3333333333333333 * (l ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0005]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(2.0 * l + N[(0.3333333333333333 * N[Power[l, 3.0], $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 5.0000000000000001e-4 < (-.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))) < 5.0000000000000001e-4

   1. Initial program 70.2%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*l* 99.9%

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

      4. pow-plus 99.9%

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

      5. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0.0005\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.0005)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (* t_0 (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* J 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.0005)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (J * 2.0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.0005)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * Math.pow(l, 2.0))) + (J * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.0005):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * math.pow(l, 2.0))) + (J * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.0005))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(J * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.0005)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l ^ 2.0))) + (J * 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]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0005]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(l * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $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 5.0000000000000001e-4 < (-.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))) < 5.0000000000000001e-4

   1. Initial program 70.2%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-5} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;t\_0 \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -1e-5) (not (<= t_0 2e-16)))
     (+ (* t_0 J) U)
     (+ U (* (cos (/ K 2.0)) (* l (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -1e-5) || !(t_0 <= 2e-16)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 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 = exp(l) - exp(-l)
    if ((t_0 <= (-1d-5)) .or. (.not. (t_0 <= 2d-16))) then
        tmp = (t_0 * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -1e-5) || !(t_0 <= 2e-16)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -1e-5) or not (t_0 <= 2e-16):
		tmp = (t_0 * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -1e-5) || !(t_0 <= 2e-16))
		tmp = Float64(Float64(t_0 * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	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 <= -1e-5) || ~((t_0 <= 2e-16)))
		tmp = (t_0 * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	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, -1e-5], N[Not[LessEqual[t$95$0, 2e-16]], $MachinePrecision]], N[(N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1.00000000000000008e-5 or 2e-16 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0 76.9%

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

   if -1.00000000000000008e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2e-16

   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 l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \cdot 10^{-5} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= t_0 -2e-170)
     (+ t_0 U)
     (+
      U
      (*
       (cos (/ K 2.0))
       (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = t_0 + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (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 = (exp(l) - exp(-l)) * j
    if (t_0 <= (-2d-170)) then
        tmp = t_0 + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = t_0 + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if t_0 <= -2e-170:
		tmp = t_0 + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (t_0 <= -2e-170)
		tmp = Float64(t_0 + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * 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 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (t_0 <= -2e-170)
		tmp = t_0 + U;
	else
		tmp = U + (cos((K / 2.0)) * (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[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-170], N[(t$95$0 + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * 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 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.99999999999999997e-170

   1. Initial program 99.4%

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

   3. Taylor expanded in K around 0 80.0%

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

   if -1.99999999999999997e-170 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 79.0%

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

   3. Taylor expanded in l around 0 95.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.13:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.092:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* J (pow l 3.0)) (* 0.3333333333333333 (cos (* K 0.5))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -9.4e+179)
     t_0
     (if (<= l -0.13)
       t_1
       (if (<= l 0.092)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* J 2.0)))))
         (if (<= l 8.4e+77) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * pow(l, 3.0)) * (0.3333333333333333 * cos((K * 0.5))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -9.4e+179) {
		tmp = t_0;
	} else if (l <= -0.13) {
		tmp = t_1;
	} else if (l <= 0.092) {
		tmp = U + (cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (J * 2.0))));
	} else if (l <= 8.4e+77) {
		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 + ((j * (l ** 3.0d0)) * (0.3333333333333333d0 * cos((k * 0.5d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-9.4d+179)) then
        tmp = t_0
    else if (l <= (-0.13d0)) then
        tmp = t_1
    else if (l <= 0.092d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * ((0.3333333333333333d0 * (j * (l ** 2.0d0))) + (j * 2.0d0))))
    else if (l <= 8.4d+77) 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 + ((J * Math.pow(l, 3.0)) * (0.3333333333333333 * Math.cos((K * 0.5))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -9.4e+179) {
		tmp = t_0;
	} else if (l <= -0.13) {
		tmp = t_1;
	} else if (l <= 0.092) {
		tmp = U + (Math.cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * Math.pow(l, 2.0))) + (J * 2.0))));
	} else if (l <= 8.4e+77) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * math.pow(l, 3.0)) * (0.3333333333333333 * math.cos((K * 0.5))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -9.4e+179:
		tmp = t_0
	elif l <= -0.13:
		tmp = t_1
	elif l <= 0.092:
		tmp = U + (math.cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * math.pow(l, 2.0))) + (J * 2.0))))
	elif l <= 8.4e+77:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * (l ^ 3.0)) * Float64(0.3333333333333333 * cos(Float64(K * 0.5)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -9.4e+179)
		tmp = t_0;
	elseif (l <= -0.13)
		tmp = t_1;
	elseif (l <= 0.092)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(J * 2.0)))));
	elseif (l <= 8.4e+77)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l ^ 3.0)) * (0.3333333333333333 * cos((K * 0.5))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -9.4e+179)
		tmp = t_0;
	elseif (l <= -0.13)
		tmp = t_1;
	elseif (l <= 0.092)
		tmp = U + (cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l ^ 2.0))) + (J * 2.0))));
	elseif (l <= 8.4e+77)
		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[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Cos[N[(K * 0.5), $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, -9.4e+179], t$95$0, If[LessEqual[l, -0.13], t$95$1, If[LessEqual[l, 0.092], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.4e+77], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.40000000000000013e179 or 8.3999999999999995e77 < 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 96.4%

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

      1. distribute-rgt-in 96.4%

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

      2. fma-define 96.4%

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

      3. associate-*l* 96.4%

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

      4. pow-plus 96.4%

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

      5. metadata-eval 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in l around inf 96.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 96.4%

         \[\leadsto \color{blue}{\left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot 0.3333333333333333} + U \]

      2. associate-*r* 96.4%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot 0.3333333333333333 + U \]

      3. *-commutative 96.4%

         \[\leadsto \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot 0.3333333333333333 + U \]

      4. associate-*l* 96.4%

         \[\leadsto \color{blue}{\left(J \cdot {\ell}^{3}\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)} + U \]

      5. *-commutative 96.4%

         \[\leadsto \left(J \cdot {\ell}^{3}\right) \cdot \left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot 0.3333333333333333\right) + U \]

   8. Simplified 96.4%

      \[\leadsto \color{blue}{\left(J \cdot {\ell}^{3}\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 0.3333333333333333\right)} + U \]

   if -9.40000000000000013e179 < l < -0.13 or 0.091999999999999998 < l < 8.3999999999999995e77

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 85.4%

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

   if -0.13 < l < 0.091999999999999998

   1. Initial program 70.2%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+179}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.13:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.092:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.00017:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* J (pow l 3.0)) (* 0.3333333333333333 (cos (* K 0.5))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -9.4e+179)
     t_0
     (if (<= l -8.5e-6)
       t_1
       (if (<= l 0.00017)
         (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
         (if (<= l 8.4e+77) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * pow(l, 3.0)) * (0.3333333333333333 * cos((K * 0.5))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -9.4e+179) {
		tmp = t_0;
	} else if (l <= -8.5e-6) {
		tmp = t_1;
	} else if (l <= 0.00017) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 8.4e+77) {
		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 + ((j * (l ** 3.0d0)) * (0.3333333333333333d0 * cos((k * 0.5d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-9.4d+179)) then
        tmp = t_0
    else if (l <= (-8.5d-6)) then
        tmp = t_1
    else if (l <= 0.00017d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    else if (l <= 8.4d+77) 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 + ((J * Math.pow(l, 3.0)) * (0.3333333333333333 * Math.cos((K * 0.5))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -9.4e+179) {
		tmp = t_0;
	} else if (l <= -8.5e-6) {
		tmp = t_1;
	} else if (l <= 0.00017) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 8.4e+77) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * math.pow(l, 3.0)) * (0.3333333333333333 * math.cos((K * 0.5))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -9.4e+179:
		tmp = t_0
	elif l <= -8.5e-6:
		tmp = t_1
	elif l <= 0.00017:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 8.4e+77:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * (l ^ 3.0)) * Float64(0.3333333333333333 * cos(Float64(K * 0.5)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -9.4e+179)
		tmp = t_0;
	elseif (l <= -8.5e-6)
		tmp = t_1;
	elseif (l <= 0.00017)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 8.4e+77)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l ^ 3.0)) * (0.3333333333333333 * cos((K * 0.5))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -9.4e+179)
		tmp = t_0;
	elseif (l <= -8.5e-6)
		tmp = t_1;
	elseif (l <= 0.00017)
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	elseif (l <= 8.4e+77)
		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[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Cos[N[(K * 0.5), $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, -9.4e+179], t$95$0, If[LessEqual[l, -8.5e-6], t$95$1, If[LessEqual[l, 0.00017], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.4e+77], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.40000000000000013e179 or 8.3999999999999995e77 < 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 96.4%

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

      1. distribute-rgt-in 96.4%

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

      2. fma-define 96.4%

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

      3. associate-*l* 96.4%

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

      4. pow-plus 96.4%

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

      5. metadata-eval 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in l around inf 96.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 96.4%

         \[\leadsto \color{blue}{\left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot 0.3333333333333333} + U \]

      2. associate-*r* 96.4%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot 0.3333333333333333 + U \]

      3. *-commutative 96.4%

         \[\leadsto \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot 0.3333333333333333 + U \]

      4. associate-*l* 96.4%

         \[\leadsto \color{blue}{\left(J \cdot {\ell}^{3}\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)} + U \]

      5. *-commutative 96.4%

         \[\leadsto \left(J \cdot {\ell}^{3}\right) \cdot \left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot 0.3333333333333333\right) + U \]

   8. Simplified 96.4%

      \[\leadsto \color{blue}{\left(J \cdot {\ell}^{3}\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot 0.3333333333333333\right)} + U \]

   if -9.40000000000000013e179 < l < -8.4999999999999999e-6 or 1.7e-4 < l < 8.3999999999999995e77

   1. Initial program 99.2%

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

   3. Taylor expanded in K around 0 85.2%

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

   if -8.4999999999999999e-6 < l < 1.7e-4

   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 l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+179}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.00017:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.01)
     (+ U (* t_0 (* l (* J 2.0))))
     (*
      U
      (+ 1.0 (/ (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U * (1.0 + ((J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))) / U));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.01d0)) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u * (1.0d0 + ((j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))) / u))
    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 (t_0 <= -0.01) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U * (1.0 + ((J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.01:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U * (1.0 + ((J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))) / U))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U * (1.0 + ((J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))) / U));
	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[t$95$0, -0.01], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 69.9%

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

      1. associate-*r* 69.9%

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

      2. *-commutative 69.9%

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

   5. Simplified 69.9%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.4%

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

   3. Taylor expanded in l around 0 90.1%

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

      1. distribute-rgt-in 90.1%

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

      2. fma-define 90.1%

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

      3. associate-*l* 90.1%

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

      4. pow-plus 90.1%

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

      5. metadata-eval 90.1%

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

   5. Simplified 90.1%

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

   6. Taylor expanded in K around 0 85.5%

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

   7. Taylor expanded in U around inf 86.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.01)
     (+ U (* t_0 (* l (* J 2.0))))
     (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (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 (t_0 <= (-0.01d0)) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (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 (t_0 <= -0.01) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.01:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(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 (t_0 <= -0.01)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (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[t$95$0, -0.01], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 85.7%

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

   3. Taylor expanded in l around 0 69.9%

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

      1. associate-*r* 69.9%

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

      2. *-commutative 69.9%

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

   5. Simplified 69.9%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.4%

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

   3. Taylor expanded in l around 0 90.1%

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

      1. distribute-rgt-in 90.1%

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

      2. fma-define 90.1%

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

      3. associate-*l* 90.1%

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

      4. pow-plus 90.1%

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

      5. metadata-eval 90.1%

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

   5. Simplified 90.1%

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

   6. Taylor expanded in K around 0 85.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+72}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+83} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.5e-6)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 1050.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (if (<= l 1.7e+72)
       (* U (- U -4.0))
       (if (or (<= l 4.3e+83) (and (not (<= l 4.5e+172)) (<= l 1.3e+194)))
         (* (* J (* l (pow K 2.0))) -0.25)
         (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 1050.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 1.7e+72) {
		tmp = U * (U - -4.0);
	} else if ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	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 <= (-8.5d-6)) then
        tmp = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    else if (l <= 1050.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 1.7d+72) then
        tmp = u * (u - (-4.0d0))
    else if ((l <= 4.3d+83) .or. (.not. (l <= 4.5d+172)) .and. (l <= 1.3d+194)) then
        tmp = (j * (l * (k ** 2.0d0))) * (-0.25d0)
    else
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 1050.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 1.7e+72) {
		tmp = U * (U - -4.0);
	} else if ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * Math.pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -8.5e-6:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	elif l <= 1050.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 1.7e+72:
		tmp = U * (U - -4.0)
	elif (l <= 4.3e+83) or (not (l <= 4.5e+172) and (l <= 1.3e+194)):
		tmp = (J * (l * math.pow(K, 2.0))) * -0.25
	else:
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.5e-6)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 1050.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 1.7e+72)
		tmp = Float64(U * Float64(U - -4.0));
	elseif ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194)))
		tmp = Float64(Float64(J * Float64(l * (K ^ 2.0))) * -0.25);
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -8.5e-6)
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	elseif (l <= 1050.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 1.7e+72)
		tmp = U * (U - -4.0);
	elseif ((l <= 4.3e+83) || (~((l <= 4.5e+172)) && (l <= 1.3e+194)))
		tmp = (J * (l * (K ^ 2.0))) * -0.25;
	else
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.5e-6], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1050.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.7e+72], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4.3e+83], And[N[Not[LessEqual[l, 4.5e+172]], $MachinePrecision], LessEqual[l, 1.3e+194]]], N[(N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -8.4999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 83.8%

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

      1. distribute-rgt-in 83.8%

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

      2. fma-define 83.8%

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

      3. associate-*l* 83.8%

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

      4. pow-plus 83.8%

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

      5. metadata-eval 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in K around 0 74.2%

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

   7. Taylor expanded in U around 0 74.1%

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

   if -8.4999999999999999e-6 < l < 1050

   1. Initial program 70.8%

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

   3. Taylor expanded in l around 0 98.3%

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

   if 1050 < l < 1.6999999999999999e72

   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 37.0%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 1.6999999999999999e72 < l < 4.3e83 or 4.5000000000000002e172 < l < 1.2999999999999999e194

   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 51.9%

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

   4. Taylor expanded in K around 0 87.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)}\right) + U \]

   5. Taylor expanded in K around inf 87.6%

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

      1. *-commutative 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if 4.3e83 < l < 4.5000000000000002e172 or 1.2999999999999999e194 < 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 95.0%

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

      1. distribute-rgt-in 95.0%

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

      2. fma-define 95.0%

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

      3. associate-*l* 95.0%

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

      4. pow-plus 95.0%

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

      5. metadata-eval 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in K around 0 73.4%

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

   7. Taylor expanded in l around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.4%

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

   9. Simplified 73.4%

      \[\leadsto U + \color{blue}{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+72}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+83} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1750:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+83} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.5e-6)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 1750.0)
     (+ U (* l (* (* J 2.0) (cos (* K 0.5)))))
     (if (<= l 3.7e+73)
       (* U (- U -4.0))
       (if (or (<= l 4.3e+83) (and (not (<= l 4.5e+172)) (<= l 1.3e+194)))
         (* (* J (* l (pow K 2.0))) -0.25)
         (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 1750.0) {
		tmp = U + (l * ((J * 2.0) * cos((K * 0.5))));
	} else if (l <= 3.7e+73) {
		tmp = U * (U - -4.0);
	} else if ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	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 <= (-8.5d-6)) then
        tmp = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    else if (l <= 1750.0d0) then
        tmp = u + (l * ((j * 2.0d0) * cos((k * 0.5d0))))
    else if (l <= 3.7d+73) then
        tmp = u * (u - (-4.0d0))
    else if ((l <= 4.3d+83) .or. (.not. (l <= 4.5d+172)) .and. (l <= 1.3d+194)) then
        tmp = (j * (l * (k ** 2.0d0))) * (-0.25d0)
    else
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 1750.0) {
		tmp = U + (l * ((J * 2.0) * Math.cos((K * 0.5))));
	} else if (l <= 3.7e+73) {
		tmp = U * (U - -4.0);
	} else if ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * Math.pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -8.5e-6:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	elif l <= 1750.0:
		tmp = U + (l * ((J * 2.0) * math.cos((K * 0.5))))
	elif l <= 3.7e+73:
		tmp = U * (U - -4.0)
	elif (l <= 4.3e+83) or (not (l <= 4.5e+172) and (l <= 1.3e+194)):
		tmp = (J * (l * math.pow(K, 2.0))) * -0.25
	else:
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.5e-6)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 1750.0)
		tmp = Float64(U + Float64(l * Float64(Float64(J * 2.0) * cos(Float64(K * 0.5)))));
	elseif (l <= 3.7e+73)
		tmp = Float64(U * Float64(U - -4.0));
	elseif ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194)))
		tmp = Float64(Float64(J * Float64(l * (K ^ 2.0))) * -0.25);
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -8.5e-6)
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	elseif (l <= 1750.0)
		tmp = U + (l * ((J * 2.0) * cos((K * 0.5))));
	elseif (l <= 3.7e+73)
		tmp = U * (U - -4.0);
	elseif ((l <= 4.3e+83) || (~((l <= 4.5e+172)) && (l <= 1.3e+194)))
		tmp = (J * (l * (K ^ 2.0))) * -0.25;
	else
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.5e-6], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1750.0], N[(U + N[(l * N[(N[(J * 2.0), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e+73], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4.3e+83], And[N[Not[LessEqual[l, 4.5e+172]], $MachinePrecision], LessEqual[l, 1.3e+194]]], N[(N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -8.4999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 83.8%

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

      1. distribute-rgt-in 83.8%

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

      2. fma-define 83.8%

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

      3. associate-*l* 83.8%

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

      4. pow-plus 83.8%

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

      5. metadata-eval 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in K around 0 74.2%

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

   7. Taylor expanded in U around 0 74.1%

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

   if -8.4999999999999999e-6 < l < 1750

   1. Initial program 70.8%

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

   3. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. associate-*l* 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-*r* 98.3%

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

      6. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   if 1750 < l < 3.69999999999999973e73

   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 37.0%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 3.69999999999999973e73 < l < 4.3e83 or 4.5000000000000002e172 < l < 1.2999999999999999e194

   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 51.9%

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

   4. Taylor expanded in K around 0 87.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)}\right) + U \]

   5. Taylor expanded in K around inf 87.6%

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

      1. *-commutative 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if 4.3e83 < l < 4.5000000000000002e172 or 1.2999999999999999e194 < 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 95.0%

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

      1. distribute-rgt-in 95.0%

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

      2. fma-define 95.0%

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

      3. associate-*l* 95.0%

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

      4. pow-plus 95.0%

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

      5. metadata-eval 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in K around 0 73.4%

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

   7. Taylor expanded in l around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.4%

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

   9. Simplified 73.4%

      \[\leadsto U + \color{blue}{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1750:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+83} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 850:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+72}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+83} \lor \neg \left(\ell \leq 2.8 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.5e-6)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 850.0)
     (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
     (if (<= l 2.05e+72)
       (* U (- U -4.0))
       (if (or (<= l 6.5e+83) (and (not (<= l 2.8e+172)) (<= l 1.3e+194)))
         (* (* J (* l (pow K 2.0))) -0.25)
         (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 850.0) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 2.05e+72) {
		tmp = U * (U - -4.0);
	} else if ((l <= 6.5e+83) || (!(l <= 2.8e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	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 <= (-8.5d-6)) then
        tmp = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    else if (l <= 850.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    else if (l <= 2.05d+72) then
        tmp = u * (u - (-4.0d0))
    else if ((l <= 6.5d+83) .or. (.not. (l <= 2.8d+172)) .and. (l <= 1.3d+194)) then
        tmp = (j * (l * (k ** 2.0d0))) * (-0.25d0)
    else
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 850.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 2.05e+72) {
		tmp = U * (U - -4.0);
	} else if ((l <= 6.5e+83) || (!(l <= 2.8e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * Math.pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -8.5e-6:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	elif l <= 850.0:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 2.05e+72:
		tmp = U * (U - -4.0)
	elif (l <= 6.5e+83) or (not (l <= 2.8e+172) and (l <= 1.3e+194)):
		tmp = (J * (l * math.pow(K, 2.0))) * -0.25
	else:
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.5e-6)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 850.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 2.05e+72)
		tmp = Float64(U * Float64(U - -4.0));
	elseif ((l <= 6.5e+83) || (!(l <= 2.8e+172) && (l <= 1.3e+194)))
		tmp = Float64(Float64(J * Float64(l * (K ^ 2.0))) * -0.25);
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -8.5e-6)
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	elseif (l <= 850.0)
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	elseif (l <= 2.05e+72)
		tmp = U * (U - -4.0);
	elseif ((l <= 6.5e+83) || (~((l <= 2.8e+172)) && (l <= 1.3e+194)))
		tmp = (J * (l * (K ^ 2.0))) * -0.25;
	else
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.5e-6], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 850.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e+72], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 6.5e+83], And[N[Not[LessEqual[l, 2.8e+172]], $MachinePrecision], LessEqual[l, 1.3e+194]]], N[(N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -8.4999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 83.8%

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

      1. distribute-rgt-in 83.8%

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

      2. fma-define 83.8%

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

      3. associate-*l* 83.8%

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

      4. pow-plus 83.8%

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

      5. metadata-eval 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in K around 0 74.2%

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

   7. Taylor expanded in U around 0 74.1%

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

   if -8.4999999999999999e-6 < l < 850

   1. Initial program 70.8%

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

   3. Taylor expanded in l around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if 850 < l < 2.04999999999999982e72

   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 37.0%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 2.04999999999999982e72 < l < 6.5000000000000003e83 or 2.8e172 < l < 1.2999999999999999e194

   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 51.9%

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

   4. Taylor expanded in K around 0 87.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)}\right) + U \]

   5. Taylor expanded in K around inf 87.6%

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

      1. *-commutative 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if 6.5000000000000003e83 < l < 2.8e172 or 1.2999999999999999e194 < 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 95.0%

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

      1. distribute-rgt-in 95.0%

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

      2. fma-define 95.0%

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

      3. associate-*l* 95.0%

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

      4. pow-plus 95.0%

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

      5. metadata-eval 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in K around 0 73.4%

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

   7. Taylor expanded in l around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.4%

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

   9. Simplified 73.4%

      \[\leadsto U + \color{blue}{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 850:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+72}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+83} \lor \neg \left(\ell \leq 2.8 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+35}:\\ \;\;\;\;U \cdot \left(1 + \frac{t\_0}{U}\right)\\ \mathbf{elif}\;\ell \leq 6600000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+83} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -1.7e+35)
     (* U (+ 1.0 (/ t_0 U)))
     (if (<= l 6600000.0)
       (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
       (if (<= l 3.55e+74)
         (* U (- U -4.0))
         (if (or (<= l 4.3e+83) (and (not (<= l 4.5e+172)) (<= l 1.3e+194)))
           (* (* J (* l (pow K 2.0))) -0.25)
           (+ U t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -1.7e+35) {
		tmp = U * (1.0 + (t_0 / U));
	} else if (l <= 6600000.0) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 3.55e+74) {
		tmp = U * (U - -4.0);
	} else if ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + 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 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-1.7d+35)) then
        tmp = u * (1.0d0 + (t_0 / u))
    else if (l <= 6600000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    else if (l <= 3.55d+74) then
        tmp = u * (u - (-4.0d0))
    else if ((l <= 4.3d+83) .or. (.not. (l <= 4.5d+172)) .and. (l <= 1.3d+194)) then
        tmp = (j * (l * (k ** 2.0d0))) * (-0.25d0)
    else
        tmp = u + t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -1.7e+35) {
		tmp = U * (1.0 + (t_0 / U));
	} else if (l <= 6600000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 3.55e+74) {
		tmp = U * (U - -4.0);
	} else if ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194))) {
		tmp = (J * (l * Math.pow(K, 2.0))) * -0.25;
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -1.7e+35:
		tmp = U * (1.0 + (t_0 / U))
	elif l <= 6600000.0:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 3.55e+74:
		tmp = U * (U - -4.0)
	elif (l <= 4.3e+83) or (not (l <= 4.5e+172) and (l <= 1.3e+194)):
		tmp = (J * (l * math.pow(K, 2.0))) * -0.25
	else:
		tmp = U + t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -1.7e+35)
		tmp = Float64(U * Float64(1.0 + Float64(t_0 / U)));
	elseif (l <= 6600000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 3.55e+74)
		tmp = Float64(U * Float64(U - -4.0));
	elseif ((l <= 4.3e+83) || (!(l <= 4.5e+172) && (l <= 1.3e+194)))
		tmp = Float64(Float64(J * Float64(l * (K ^ 2.0))) * -0.25);
	else
		tmp = Float64(U + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -1.7e+35)
		tmp = U * (1.0 + (t_0 / U));
	elseif (l <= 6600000.0)
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	elseif (l <= 3.55e+74)
		tmp = U * (U - -4.0);
	elseif ((l <= 4.3e+83) || (~((l <= 4.5e+172)) && (l <= 1.3e+194)))
		tmp = (J * (l * (K ^ 2.0))) * -0.25;
	else
		tmp = U + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.7e+35], N[(U * N[(1.0 + N[(t$95$0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6600000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.55e+74], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4.3e+83], And[N[Not[LessEqual[l, 4.5e+172]], $MachinePrecision], LessEqual[l, 1.3e+194]]], N[(N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], N[(U + t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.7000000000000001e35

   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 88.8%

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

      1. distribute-rgt-in 88.8%

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

      2. fma-define 88.8%

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

      3. associate-*l* 88.8%

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

      4. pow-plus 88.8%

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

      5. metadata-eval 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in K around 0 78.5%

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

   7. Taylor expanded in U around inf 83.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}{U}\right)} \]

   8. Taylor expanded in l around inf 83.3%

      \[\leadsto U \cdot \left(1 + \frac{\color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)}}{U}\right) \]
   9. Step-by-step derivation

      1. *-commutative 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*r* 78.5%

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

   10. Simplified 83.3%

       \[\leadsto U \cdot \left(1 + \frac{\color{blue}{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)}}{U}\right) \]

   if -1.7000000000000001e35 < l < 6.6e6

   1. Initial program 71.7%

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

   3. Taylor expanded in l around 0 95.5%

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

      1. associate-*r* 95.5%

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

      2. *-commutative 95.5%

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

   5. Simplified 95.5%

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

   if 6.6e6 < l < 3.55000000000000001e74

   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 37.0%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 3.55000000000000001e74 < l < 4.3e83 or 4.5000000000000002e172 < l < 1.2999999999999999e194

   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 51.9%

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

   4. Taylor expanded in K around 0 87.6%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)}\right) + U \]

   5. Taylor expanded in K around inf 87.6%

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

      1. *-commutative 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if 4.3e83 < l < 4.5000000000000002e172 or 1.2999999999999999e194 < 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 95.0%

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

      1. distribute-rgt-in 95.0%

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

      2. fma-define 95.0%

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

      3. associate-*l* 95.0%

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

      4. pow-plus 95.0%

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

      5. metadata-eval 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in K around 0 73.4%

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

   7. Taylor expanded in l around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.4%

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

   9. Simplified 73.4%

      \[\leadsto U + \color{blue}{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+35}:\\ \;\;\;\;U \cdot \left(1 + \frac{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)}{U}\right)\\ \mathbf{elif}\;\ell \leq 6600000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+83} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+172}\right) \land \ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+32}:\\ \;\;\;\;U \cdot \left(1 + \frac{t\_0}{U}\right)\\ \mathbf{elif}\;\ell \leq 0.000175:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+208}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -2.15e+32)
     (* U (+ 1.0 (/ t_0 U)))
     (if (<= l 0.000175)
       (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
       (if (<= l 1.45e+208)
         (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0)))))))
         (+ U t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -2.15e+32) {
		tmp = U * (1.0 + (t_0 / U));
	} else if (l <= 0.000175) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 1.45e+208) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	} else {
		tmp = U + 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 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-2.15d+32)) then
        tmp = u * (1.0d0 + (t_0 / u))
    else if (l <= 0.000175d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    else if (l <= 1.45d+208) then
        tmp = u + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k ** 2.0d0))))))
    else
        tmp = u + t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -2.15e+32) {
		tmp = U * (1.0 + (t_0 / U));
	} else if (l <= 0.000175) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 1.45e+208) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0))))));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -2.15e+32:
		tmp = U * (1.0 + (t_0 / U))
	elif l <= 0.000175:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 1.45e+208:
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	else:
		tmp = U + t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -2.15e+32)
		tmp = Float64(U * Float64(1.0 + Float64(t_0 / U)));
	elseif (l <= 0.000175)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 1.45e+208)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))));
	else
		tmp = Float64(U + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -2.15e+32)
		tmp = U * (1.0 + (t_0 / U));
	elseif (l <= 0.000175)
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	elseif (l <= 1.45e+208)
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K ^ 2.0))))));
	else
		tmp = U + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.15e+32], N[(U * N[(1.0 + N[(t$95$0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.000175], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+208], N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.1499999999999999e32

   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 88.8%

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

      1. distribute-rgt-in 88.8%

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

      2. fma-define 88.8%

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

      3. associate-*l* 88.8%

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

      4. pow-plus 88.8%

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

      5. metadata-eval 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in K around 0 78.5%

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

   7. Taylor expanded in U around inf 83.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}{U}\right)} \]

   8. Taylor expanded in l around inf 83.3%

      \[\leadsto U \cdot \left(1 + \frac{\color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)}}{U}\right) \]
   9. Step-by-step derivation

      1. *-commutative 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*r* 78.5%

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

   10. Simplified 83.3%

       \[\leadsto U \cdot \left(1 + \frac{\color{blue}{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)}}{U}\right) \]

   if -2.1499999999999999e32 < l < 1.74999999999999998e-4

   1. Initial program 71.0%

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

   3. Taylor expanded in l around 0 96.9%

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

      1. associate-*r* 96.9%

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

      2. *-commutative 96.9%

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

   5. Simplified 96.9%

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

   if 1.74999999999999998e-4 < l < 1.45000000000000004e208

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 25.5%

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

   4. Taylor expanded in K around 0 49.1%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + -0.125 \cdot \left({K}^{2} \cdot \ell\right)\right)}\right) + U \]

   if 1.45000000000000004e208 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. fma-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 89.5%

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

   7. Taylor expanded in l around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*r* 89.5%

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

   9. Simplified 89.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+32}:\\ \;\;\;\;U \cdot \left(1 + \frac{{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)}{U}\right)\\ \mathbf{elif}\;\ell \leq 0.000175:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+208}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9500000000000:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1650:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -9500000000000.0)
   (* J (* 0.3333333333333333 (pow l 3.0)))
   (if (<= l 1650.0)
     (fma J (* l 2.0) U)
     (if (<= l 1.3e+146)
       (* U (- U -4.0))
       (+ U (* (pow l 3.0) (* J 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -9500000000000.0) {
		tmp = J * (0.3333333333333333 * pow(l, 3.0));
	} else if (l <= 1650.0) {
		tmp = fma(J, (l * 2.0), U);
	} else if (l <= 1.3e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -9500000000000.0)
		tmp = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)));
	elseif (l <= 1650.0)
		tmp = fma(J, Float64(l * 2.0), U);
	elseif (l <= 1.3e+146)
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -9500000000000.0], N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1650.0], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.3e+146], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.5e12

   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. Step-by-step derivation

      1. distribute-rgt-in 86.0%

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

      2. fma-define 86.0%

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

      3. associate-*l* 86.0%

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

      4. pow-plus 86.0%

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

      5. metadata-eval 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in K around 0 76.0%

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

   7. Taylor expanded in U around inf 80.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}{U}\right)} \]

   8. Taylor expanded in l around inf 76.0%

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

      1. associate-*r* 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   10. Simplified 76.0%

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

   if -9.5e12 < l < 1650

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0 96.8%

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

   4. Taylor expanded in K around 0 83.2%

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

      1. +-commutative 83.2%

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

      2. *-commutative 83.2%

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

      3. associate-*l* 83.2%

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

      4. *-commutative 83.2%

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

      5. fma-define 83.2%

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

      6. *-commutative 83.2%

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

   6. Simplified 83.2%

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

   if 1650 < l < 1.30000000000000007e146

   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 41.7%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 1.30000000000000007e146 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. fma-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 75.0%

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

   7. Taylor expanded in l around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*r* 75.0%

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

   9. Simplified 75.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9500000000000:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1650:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 980000:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.5e-6)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 980000.0)
     (fma J (* l 2.0) U)
     (if (<= l 1.3e+146)
       (* U (- U -4.0))
       (+ U (* (pow l 3.0) (* J 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e-6) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 980000.0) {
		tmp = fma(J, (l * 2.0), U);
	} else if (l <= 1.3e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.5e-6)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 980000.0)
		tmp = fma(J, Float64(l * 2.0), U);
	elseif (l <= 1.3e+146)
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.5e-6], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 980000.0], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.3e+146], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.4999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 83.8%

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

      1. distribute-rgt-in 83.8%

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

      2. fma-define 83.8%

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

      3. associate-*l* 83.8%

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

      4. pow-plus 83.8%

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

      5. metadata-eval 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in K around 0 74.2%

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

   7. Taylor expanded in U around 0 74.1%

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

   if -8.4999999999999999e-6 < l < 9.8e5

   1. Initial program 70.8%

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

   3. Taylor expanded in l around 0 98.3%

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

   4. Taylor expanded in K around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-*l* 84.3%

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

      4. *-commutative 84.3%

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

      5. fma-define 84.4%

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

      6. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   if 9.8e5 < l < 1.30000000000000007e146

   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 41.7%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 1.30000000000000007e146 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. fma-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 75.0%

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

   7. Taylor expanded in l around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*r* 75.0%

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

   9. Simplified 75.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 980000:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -650000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 0.3333333333333333 (pow l 3.0)))))
   (if (<= l -650000000000.0)
     t_0
     (if (<= l 1050.0)
       (fma J (* l 2.0) U)
       (if (<= l 1.3e+146) (* U (- U -4.0)) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (0.3333333333333333 * pow(l, 3.0));
	double tmp;
	if (l <= -650000000000.0) {
		tmp = t_0;
	} else if (l <= 1050.0) {
		tmp = fma(J, (l * 2.0), U);
	} else if (l <= 1.3e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -650000000000.0)
		tmp = t_0;
	elseif (l <= 1050.0)
		tmp = fma(J, Float64(l * 2.0), U);
	elseif (l <= 1.3e+146)
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -650000000000.0], t$95$0, If[LessEqual[l, 1050.0], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.3e+146], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5e11 or 1.30000000000000007e146 < 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 90.8%

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

      1. distribute-rgt-in 90.8%

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

      2. fma-define 90.8%

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

      3. associate-*l* 90.8%

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

      4. pow-plus 90.8%

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

      5. metadata-eval 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in K around 0 75.6%

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

   7. Taylor expanded in U around inf 78.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}{U}\right)} \]

   8. Taylor expanded in l around inf 75.7%

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

      1. associate-*r* 75.7%

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

      2. *-commutative 75.7%

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

      3. associate-*r* 75.7%

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

   10. Simplified 75.7%

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

   if -6.5e11 < l < 1050

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0 96.8%

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

   4. Taylor expanded in K around 0 83.2%

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

      1. +-commutative 83.2%

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

      2. *-commutative 83.2%

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

      3. associate-*l* 83.2%

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

      4. *-commutative 83.2%

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

      5. fma-define 83.2%

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

      6. *-commutative 83.2%

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

   6. Simplified 83.2%

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

   if 1050 < l < 1.30000000000000007e146

   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 41.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -650000000000:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.6% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2050000000 \lor \neg \left(\ell \leq 1.3 \cdot 10^{+146}\right):\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l 2050000000.0) (not (<= l 1.3e+146)))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (* U (- U -4.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 2050000000.0) || !(l <= 1.3e+146)) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= 2050000000.0d0) .or. (.not. (l <= 1.3d+146))) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u * (u - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 2050000000.0) || !(l <= 1.3e+146)) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= 2050000000.0) or not (l <= 1.3e+146):
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U * (U - -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= 2050000000.0) || !(l <= 1.3e+146))
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U * Float64(U - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= 2050000000.0) || ~((l <= 1.3e+146)))
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U * (U - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, 2050000000.0], N[Not[LessEqual[l, 1.3e+146]], $MachinePrecision]], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.05e9 or 1.30000000000000007e146 < l

   1. Initial program 82.9%

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

   3. Taylor expanded in l around 0 94.7%

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

      1. distribute-rgt-in 94.7%

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

      2. fma-define 94.7%

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

      3. associate-*l* 94.7%

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

      4. pow-plus 94.7%

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

      5. metadata-eval 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in K around 0 80.4%

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

   7. Taylor expanded in U around inf 80.8%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}{U}\right)} \]

   8. Taylor expanded in l around 0 64.6%

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

      1. associate-/l* 68.8%

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

   10. Simplified 68.8%

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

   if 2.05e9 < l < 1.30000000000000007e146

   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 41.7%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2050000000 \lor \neg \left(\ell \leq 1.3 \cdot 10^{+146}\right):\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.3% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 500:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+146}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* l J))))
   (if (<= l -3.1e-9)
     t_0
     (if (<= l 500.0) U (if (<= l 1.8e+146) (* U U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double tmp;
	if (l <= -3.1e-9) {
		tmp = t_0;
	} else if (l <= 500.0) {
		tmp = U;
	} else if (l <= 1.8e+146) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (l * j)
    if (l <= (-3.1d-9)) then
        tmp = t_0
    else if (l <= 500.0d0) then
        tmp = u
    else if (l <= 1.8d+146) then
        tmp = u * u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double tmp;
	if (l <= -3.1e-9) {
		tmp = t_0;
	} else if (l <= 500.0) {
		tmp = U;
	} else if (l <= 1.8e+146) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (l * J)
	tmp = 0
	if l <= -3.1e-9:
		tmp = t_0
	elif l <= 500.0:
		tmp = U
	elif l <= 1.8e+146:
		tmp = U * U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(l * J))
	tmp = 0.0
	if (l <= -3.1e-9)
		tmp = t_0;
	elseif (l <= 500.0)
		tmp = U;
	elseif (l <= 1.8e+146)
		tmp = Float64(U * U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (l * J);
	tmp = 0.0;
	if (l <= -3.1e-9)
		tmp = t_0;
	elseif (l <= 500.0)
		tmp = U;
	elseif (l <= 1.8e+146)
		tmp = U * U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.1e-9], t$95$0, If[LessEqual[l, 500.0], U, If[LessEqual[l, 1.8e+146], N[(U * U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.10000000000000005e-9 or 1.7999999999999999e146 < l

   1. Initial program 99.4%

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

   3. Taylor expanded in l around 0 39.6%

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

   4. Taylor expanded in J around inf 43.4%

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

   5. Taylor expanded in J around inf 39.7%

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

      1. associate-*r* 39.7%

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

   7. Simplified 39.7%

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

   8. Taylor expanded in K around 0 32.1%

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

   if -3.10000000000000005e-9 < l < 500

   1. Initial program 70.9%

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

   3. Taylor expanded in J around 0 68.8%

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

   if 500 < l < 1.7999999999999999e146

   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 41.6%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 500:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+146}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.3% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 760:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 2.0 (* l J))))
   (if (<= l -3.1e-9)
     t_0
     (if (<= l 760.0) U (if (<= l 1.3e+146) (* U (- U -4.0)) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double tmp;
	if (l <= -3.1e-9) {
		tmp = t_0;
	} else if (l <= 760.0) {
		tmp = U;
	} else if (l <= 1.3e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (l * j)
    if (l <= (-3.1d-9)) then
        tmp = t_0
    else if (l <= 760.0d0) then
        tmp = u
    else if (l <= 1.3d+146) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 * (l * J);
	double tmp;
	if (l <= -3.1e-9) {
		tmp = t_0;
	} else if (l <= 760.0) {
		tmp = U;
	} else if (l <= 1.3e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 * (l * J)
	tmp = 0
	if l <= -3.1e-9:
		tmp = t_0
	elif l <= 760.0:
		tmp = U
	elif l <= 1.3e+146:
		tmp = U * (U - -4.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 * Float64(l * J))
	tmp = 0.0
	if (l <= -3.1e-9)
		tmp = t_0;
	elseif (l <= 760.0)
		tmp = U;
	elseif (l <= 1.3e+146)
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 * (l * J);
	tmp = 0.0;
	if (l <= -3.1e-9)
		tmp = t_0;
	elseif (l <= 760.0)
		tmp = U;
	elseif (l <= 1.3e+146)
		tmp = U * (U - -4.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.1e-9], t$95$0, If[LessEqual[l, 760.0], U, If[LessEqual[l, 1.3e+146], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.10000000000000005e-9 or 1.30000000000000007e146 < l

   1. Initial program 99.4%

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

   3. Taylor expanded in l around 0 39.6%

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

   4. Taylor expanded in J around inf 43.4%

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

   5. Taylor expanded in J around inf 39.7%

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

      1. associate-*r* 39.7%

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

   7. Simplified 39.7%

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

   8. Taylor expanded in K around 0 32.1%

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

   if -3.10000000000000005e-9 < l < 760

   1. Initial program 70.9%

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

   3. Taylor expanded in J around 0 68.8%

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

   if 760 < l < 1.30000000000000007e146

   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 41.7%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 760:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 520:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 520.0)
   (+ U (* J (* l 2.0)))
   (if (<= l 1.55e+146) (* U (- U -4.0)) (* J (+ (* l 2.0) (/ U J))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 520.0) {
		tmp = U + (J * (l * 2.0));
	} else if (l <= 1.55e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 520.0d0) then
        tmp = u + (j * (l * 2.0d0))
    else if (l <= 1.55d+146) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = j * ((l * 2.0d0) + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 520.0) {
		tmp = U + (J * (l * 2.0));
	} else if (l <= 1.55e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 520.0:
		tmp = U + (J * (l * 2.0))
	elif l <= 1.55e+146:
		tmp = U * (U - -4.0)
	else:
		tmp = J * ((l * 2.0) + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 520.0)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	elseif (l <= 1.55e+146)
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 520.0)
		tmp = U + (J * (l * 2.0));
	elseif (l <= 1.55e+146)
		tmp = U * (U - -4.0);
	else
		tmp = J * ((l * 2.0) + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 520.0], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+146], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 520

   1. Initial program 80.2%

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

   3. Taylor expanded in l around 0 78.4%

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

   4. Taylor expanded in K around 0 66.9%

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

      1. +-commutative 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

   if 520 < l < 1.5500000000000001e146

   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 41.7%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 1.5500000000000001e146 < 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 43.0%

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

   4. Taylor expanded in J around inf 48.8%

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

   5. Taylor expanded in K around 0 39.3%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 520:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 54.3% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1100:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1100.0)
   (+ U (* J (* l 2.0)))
   (if (<= l 1.45e+146) (* U (- U -4.0)) (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1100.0) {
		tmp = U + (J * (l * 2.0));
	} else if (l <= 1.45e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = 2.0 * (l * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 1100.0d0) then
        tmp = u + (j * (l * 2.0d0))
    else if (l <= 1.45d+146) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = 2.0d0 * (l * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1100.0) {
		tmp = U + (J * (l * 2.0));
	} else if (l <= 1.45e+146) {
		tmp = U * (U - -4.0);
	} else {
		tmp = 2.0 * (l * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1100.0:
		tmp = U + (J * (l * 2.0))
	elif l <= 1.45e+146:
		tmp = U * (U - -4.0)
	else:
		tmp = 2.0 * (l * J)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1100.0)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	elseif (l <= 1.45e+146)
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = Float64(2.0 * Float64(l * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1100.0)
		tmp = U + (J * (l * 2.0));
	elseif (l <= 1.45e+146)
		tmp = U * (U - -4.0);
	else
		tmp = 2.0 * (l * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1100.0], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+146], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1100

   1. Initial program 80.2%

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

   3. Taylor expanded in l around 0 78.4%

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

   4. Taylor expanded in K around 0 66.9%

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

      1. +-commutative 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

   if 1100 < l < 1.4499999999999999e146

   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 41.7%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if 1.4499999999999999e146 < 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 43.0%

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

   4. Taylor expanded in J around inf 48.8%

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

   5. Taylor expanded in J around inf 43.3%

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

      1. associate-*r* 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in K around 0 33.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1100:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1020000000 \lor \neg \left(\ell \leq 14000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1020000000.0) (not (<= l 14000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1020000000.0) || !(l <= 14000.0)) {
		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 <= (-1020000000.0d0)) .or. (.not. (l <= 14000.0d0))) 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 <= -1020000000.0) || !(l <= 14000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1020000000.0) or not (l <= 14000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1020000000.0) || !(l <= 14000.0))
		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 <= -1020000000.0) || ~((l <= 14000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1020000000.0], N[Not[LessEqual[l, 14000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.02e9 or 14000 < 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 20.2%

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

   if -1.02e9 < l < 14000

   1. Initial program 71.1%

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

   3. Taylor expanded in J around 0 67.3%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1020000000 \lor \neg \left(\ell \leq 14000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 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 2.9%

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

   1. *-inverses 2.9%

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

6. Simplified 2.9%

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

7. Final simplification 2.9%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 24: 36.9% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 36.6%

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

4. Final simplification 36.6%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))