Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.6%

Time: 18.5s

Alternatives: 18

Speedup: 1.4×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -100000:\\ \;\;\;\;t_0 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot t_1, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_1 \cdot \left(t_0 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 -100000.0)
     (+ (* t_0 (* J (cos (* 0.5 K)))) U)
     (if (<= t_0 0.0) (fma J (* (* l 2.0) t_1) U) (+ U (* t_1 (* t_0 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = (t_0 * (J * cos((0.5 * K)))) + U;
	} else if (t_0 <= 0.0) {
		tmp = fma(J, ((l * 2.0) * t_1), U);
	} else {
		tmp = U + (t_1 * (t_0 * J));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -100000.0)
		tmp = Float64(Float64(t_0 * Float64(J * cos(Float64(0.5 * K)))) + U);
	elseif (t_0 <= 0.0)
		tmp = fma(J, Float64(Float64(l * 2.0) * t_1), U);
	else
		tmp = Float64(U + Float64(t_1 * Float64(t_0 * J)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -100000.0], N[(N[(t$95$0 * N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(J * N[(N[(l * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$1 * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e5

   1. Initial program 100.0%

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

   2. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -1e5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. fma-def 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in l around 0 99.9%

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

   if 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -100000:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -100000 \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;U + t_0 \cdot \left(t_1 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot t_0, U\right)\\ \end{array} \end{array} \]

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 -100000.0) (not (<= t_1 0.0)))
     (+ U (* t_0 (* t_1 J)))
     (fma J (* (* l 2.0) t_0) U))))

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 <= -100000.0) || !(t_1 <= 0.0)) {
		tmp = U + (t_0 * (t_1 * J));
	} else {
		tmp = fma(J, ((l * 2.0) * t_0), U);
	}
	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 <= -100000.0) || !(t_1 <= 0.0))
		tmp = Float64(U + Float64(t_0 * Float64(t_1 * J)));
	else
		tmp = fma(J, Float64(Float64(l * 2.0) * t_0), U);
	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, -100000.0], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e5 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -1e5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. fma-def 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -100000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]

Alternative 3: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -100000 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;U + t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -100000.0) (not (<= t_0 0.0)))
     (+ U (* t_0 J))
     (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -100000.0) || !(t_0 <= 0.0)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	}
	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 <= (-100000.0d0)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = u + (t_0 * j)
    else
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    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 <= -100000.0) || !(t_0 <= 0.0)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -100000.0) or not (t_0 <= 0.0):
		tmp = U + (t_0 * J)
	else:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -100000.0) || !(t_0 <= 0.0))
		tmp = Float64(U + Float64(t_0 * J));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	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 <= -100000.0) || ~((t_0 <= 0.0)))
		tmp = U + (t_0 * J);
	else
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	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, -100000.0], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(U + N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e5 or 0.0 < (-.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. Taylor expanded in K around 0 74.6%

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

   if -1e5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 71.3%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -100000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 4: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.92:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.02:\\ \;\;\;\;U + \left(J \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.92)
     (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))
     (if (<= t_0 -0.02)
       (+ U (* (* J (* K K)) (* l -0.25)))
       (+ U (* J (+ (* l 2.0) (* 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 tmp;
	if (t_0 <= -0.92) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else if (t_0 <= -0.02) {
		tmp = U + ((J * (K * K)) * (l * -0.25));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.92d0)) then
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    else if (t_0 <= (-0.02d0)) then
        tmp = u + ((j * (k * k)) * (l * (-0.25d0)))
    else
        tmp = u + (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.92) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else if (t_0 <= -0.02) {
		tmp = U + ((J * (K * K)) * (l * -0.25));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.92:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	elif t_0 <= -0.02:
		tmp = U + ((J * (K * K)) * (l * -0.25))
	else:
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.92)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	elseif (t_0 <= -0.02)
		tmp = Float64(U + Float64(Float64(J * Float64(K * K)) * Float64(l * -0.25)));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))));
	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 <= -0.92)
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	elseif (t_0 <= -0.02)
		tmp = U + ((J * (K * K)) * (l * -0.25));
	else
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0))));
	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, -0.92], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(U + N[(N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(l * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.92000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 61.2%

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

   if -0.92000000000000004 < (cos.f64 (/.f64 K 2)) < -0.0200000000000000004

   1. Initial program 88.2%

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

   2. Taylor expanded in l around 0 55.9%

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

   3. Taylor expanded in K around 0 55.0%

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

      1. associate-*r* 55.0%

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

      2. associate-*r* 55.0%

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

      3. associate-*r* 55.0%

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

      4. distribute-rgt-out 69.7%

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

      5. *-commutative 69.7%

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

      6. associate-*r* 69.7%

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

      7. distribute-rgt-out 69.7%

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

      8. unpow2 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in K around inf 69.7%

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

      1. *-commutative 69.7%

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

      2. associate-*r* 69.7%

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

      3. associate-*l* 69.7%

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

      4. unpow2 69.7%

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

   8. Simplified 69.7%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 84.7%

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

   2. Taylor expanded in J around 0 84.7%

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

      1. associate-*r* 84.7%

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

      2. *-commutative 84.7%

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

   4. Simplified 84.7%

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

   5. Taylor expanded in K around 0 84.0%

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

   6. Taylor expanded in l around 0 81.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.92:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(J \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 5: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+176}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -3.3 \lor \neg \left(\ell \leq 3 \cdot 10^{-14}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5e+176)
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (if (or (<= l -3.3) (not (<= l 3e-14)))
     (+ U (* (- (exp l) (exp (- l))) (+ J (* J (* K (* K -0.125))))))
     (fma J (* (* l 2.0) (cos (/ K 2.0))) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5e+176) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else if ((l <= -3.3) || !(l <= 3e-14)) {
		tmp = U + ((exp(l) - exp(-l)) * (J + (J * (K * (K * -0.125)))));
	} else {
		tmp = fma(J, ((l * 2.0) * cos((K / 2.0))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5e+176)
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	elseif ((l <= -3.3) || !(l <= 3e-14))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * Float64(J + Float64(J * Float64(K * Float64(K * -0.125))))));
	else
		tmp = fma(J, Float64(Float64(l * 2.0) * cos(Float64(K / 2.0))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5e+176], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -3.3], N[Not[LessEqual[l, 3e-14]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * N[(J + N[(J * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e176

   1. Initial program 100.0%

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

   2. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in K around 0 95.8%

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

   6. Taylor expanded in l around 0 95.8%

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

   7. Taylor expanded in l around inf 95.8%

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

   if -5e176 < l < -3.2999999999999998 or 2.9999999999999998e-14 < 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. Taylor expanded in K around 0 3.5%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(J \cdot \left({K}^{2} \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\right) + J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} + U \]
   3. Step-by-step derivation

      1. +-commutative 3.5%

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

      2. associate-*r* 3.5%

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

      3. associate-*r* 3.5%

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

      4. distribute-rgt-out 81.9%

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

      5. associate-*r* 81.9%

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

      6. *-commutative 81.9%

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

      7. associate-*l* 81.9%

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

      8. *-commutative 81.9%

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

      9. unpow2 81.9%

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

      10. associate-*l* 81.9%

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

   4. Simplified 81.9%

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

   if -3.2999999999999998 < l < 2.9999999999999998e-14

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. fma-def 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in l around 0 99.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+176}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -3.3 \lor \neg \left(\ell \leq 3 \cdot 10^{-14}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]

Alternative 6: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0009 \lor \neg \left(\ell \leq 2.3 \cdot 10^{-31}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0009) (not (<= l 2.3e-31)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (fma J (* (* l 2.0) (cos (/ K 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0009) || !(l <= 2.3e-31)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = fma(J, ((l * 2.0) * cos((K / 2.0))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.0009) || !(l <= 2.3e-31))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = fma(J, Float64(Float64(l * 2.0) * cos(Float64(K / 2.0))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0009], N[Not[LessEqual[l, 2.3e-31]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.9999999999999998e-4 or 2.2999999999999998e-31 < 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. Taylor expanded in K around 0 75.0%

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

   if -8.9999999999999998e-4 < l < 2.2999999999999998e-31

   1. Initial program 70.9%

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

      1. associate-*l* 70.9%

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

      2. fma-def 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in l around 0 99.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0009 \lor \neg \left(\ell \leq 2.3 \cdot 10^{-31}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(\ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]

Alternative 7: 78.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{+20}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq -1300000000:\\ \;\;\;\;U + J \cdot \left(512 + K \cdot \left(K \cdot -64\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-31}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -1.55e+55)
     t_0
     (if (<= l -7.5e+20)
       (+ U (* l (* J (+ 2.0 (* (* K K) -0.25)))))
       (if (<= l -1300000000.0)
         (+ U (* J (+ 512.0 (* K (* K -64.0)))))
         (if (<= l 2.3e-31) (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -1.55e+55) {
		tmp = t_0;
	} else if (l <= -7.5e+20) {
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	} else if (l <= -1300000000.0) {
		tmp = U + (J * (512.0 + (K * (K * -64.0))));
	} else if (l <= 2.3e-31) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-1.55d+55)) then
        tmp = t_0
    else if (l <= (-7.5d+20)) then
        tmp = u + (l * (j * (2.0d0 + ((k * k) * (-0.25d0)))))
    else if (l <= (-1300000000.0d0)) then
        tmp = u + (j * (512.0d0 + (k * (k * (-64.0d0)))))
    else if (l <= 2.3d-31) then
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    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 + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -1.55e+55) {
		tmp = t_0;
	} else if (l <= -7.5e+20) {
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	} else if (l <= -1300000000.0) {
		tmp = U + (J * (512.0 + (K * (K * -64.0))));
	} else if (l <= 2.3e-31) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -1.55e+55:
		tmp = t_0
	elif l <= -7.5e+20:
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))))
	elif l <= -1300000000.0:
		tmp = U + (J * (512.0 + (K * (K * -64.0))))
	elif l <= 2.3e-31:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -1.55e+55)
		tmp = t_0;
	elseif (l <= -7.5e+20)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(Float64(K * K) * -0.25)))));
	elseif (l <= -1300000000.0)
		tmp = Float64(U + Float64(J * Float64(512.0 + Float64(K * Float64(K * -64.0)))));
	elseif (l <= 2.3e-31)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -1.55e+55)
		tmp = t_0;
	elseif (l <= -7.5e+20)
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	elseif (l <= -1300000000.0)
		tmp = U + (J * (512.0 + (K * (K * -64.0))));
	elseif (l <= 2.3e-31)
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.55e+55], t$95$0, If[LessEqual[l, -7.5e+20], N[(U + N[(l * N[(J * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1300000000.0], N[(U + N[(J * N[(512.0 + N[(K * N[(K * -64.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-31], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.54999999999999997e55 or 2.2999999999999998e-31 < 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. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in K around 0 77.8%

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

   6. Taylor expanded in l around 0 62.7%

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

   7. Taylor expanded in l around inf 62.7%

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

   if -1.54999999999999997e55 < l < -7.5e20

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 2.5%

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

   3. Taylor expanded in K around 0 38.9%

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

      1. associate-*r* 38.9%

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

      2. associate-*r* 38.9%

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

      3. associate-*r* 38.9%

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

      4. distribute-rgt-out 38.9%

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

      5. *-commutative 38.9%

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

      6. associate-*r* 38.9%

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

      7. distribute-rgt-out 38.9%

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

      8. unpow2 38.9%

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

   5. Simplified 38.9%

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

   if -7.5e20 < l < -1.3e9

   1. Initial program 100.0%

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

   3. Applied egg-rr 3.8%

      \[\leadsto \left(J \cdot \color{blue}{512}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 68.4%

      \[\leadsto \color{blue}{\left(-64 \cdot \left(J \cdot {K}^{2}\right) + 512 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. +-commutative 68.4%

         \[\leadsto \color{blue}{\left(512 \cdot J + -64 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]

      2. *-commutative 68.4%

         \[\leadsto \left(512 \cdot J + -64 \cdot \color{blue}{\left({K}^{2} \cdot J\right)}\right) + U \]

      3. associate-*r* 68.4%

         \[\leadsto \left(512 \cdot J + \color{blue}{\left(-64 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      4. distribute-rgt-out 68.4%

         \[\leadsto \color{blue}{J \cdot \left(512 + -64 \cdot {K}^{2}\right)} + U \]

      5. *-commutative 68.4%

         \[\leadsto J \cdot \left(512 + \color{blue}{{K}^{2} \cdot -64}\right) + U \]

      6. unpow2 68.4%

         \[\leadsto J \cdot \left(512 + \color{blue}{\left(K \cdot K\right)} \cdot -64\right) + U \]

      7. associate-*l* 68.4%

         \[\leadsto J \cdot \left(512 + \color{blue}{K \cdot \left(K \cdot -64\right)}\right) + U \]

   6. Simplified 68.4%

      \[\leadsto \color{blue}{J \cdot \left(512 + K \cdot \left(K \cdot -64\right)\right)} + U \]

   if -1.3e9 < l < 2.2999999999999998e-31

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0 95.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+55}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -7.5 \cdot 10^{+20}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq -1300000000:\\ \;\;\;\;U + J \cdot \left(512 + K \cdot \left(K \cdot -64\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-31}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 8: 72.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.46 \cdot 10^{+20}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -1.6e+55)
     t_0
     (if (<= l -1.46e+20)
       (+ U (* l (* J (+ 2.0 (* (* K K) -0.25)))))
       (if (<= l 2.3e-31) (fma (* J 2.0) l U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -1.6e+55) {
		tmp = t_0;
	} else if (l <= -1.46e+20) {
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	} else if (l <= 2.3e-31) {
		tmp = fma((J * 2.0), l, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -1.6e+55)
		tmp = t_0;
	elseif (l <= -1.46e+20)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(Float64(K * K) * -0.25)))));
	elseif (l <= 2.3e-31)
		tmp = fma(Float64(J * 2.0), l, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.6e+55], t$95$0, If[LessEqual[l, -1.46e+20], N[(U + N[(l * N[(J * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-31], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.6000000000000001e55 or 2.2999999999999998e-31 < 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. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in K around 0 77.8%

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

   6. Taylor expanded in l around 0 62.7%

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

   7. Taylor expanded in l around inf 62.7%

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

   if -1.6000000000000001e55 < l < -1.46e20

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 2.5%

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

   3. Taylor expanded in K around 0 38.9%

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

      1. associate-*r* 38.9%

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

      2. associate-*r* 38.9%

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

      3. associate-*r* 38.9%

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

      4. distribute-rgt-out 38.9%

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

      5. *-commutative 38.9%

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

      6. associate-*r* 38.9%

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

      7. distribute-rgt-out 38.9%

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

      8. unpow2 38.9%

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

   5. Simplified 38.9%

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

   if -1.46e20 < l < 2.2999999999999998e-31

   1. Initial program 72.9%

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

   2. Taylor expanded in l around 0 93.3%

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

   3. Taylor expanded in K around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. associate-*r* 84.5%

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

      3. fma-def 84.5%

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

      4. *-commutative 84.5%

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

   5. Simplified 84.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{+55}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -1.46 \cdot 10^{+20}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 9: 56.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.4 \cdot 10^{+215}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{+152}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1100:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.4e+215)
   (+ U (* 2.0 (* l J)))
   (if (<= l -6e+152)
     (* U U)
     (if (<= l -1100.0)
       (- -8.0 (* U U))
       (if (<= l 2.45e+27)
         (fma (* J 2.0) l U)
         (+ U (* l (* J (+ 2.0 (* (* K K) -0.25))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.4e+215) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= -6e+152) {
		tmp = U * U;
	} else if (l <= -1100.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 2.45e+27) {
		tmp = fma((J * 2.0), l, U);
	} else {
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.4e+215)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	elseif (l <= -6e+152)
		tmp = Float64(U * U);
	elseif (l <= -1100.0)
		tmp = Float64(-8.0 - Float64(U * U));
	elseif (l <= 2.45e+27)
		tmp = fma(Float64(J * 2.0), l, U);
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(Float64(K * K) * -0.25)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.4e+215], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -6e+152], N[(U * U), $MachinePrecision], If[LessEqual[l, -1100.0], N[(-8.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.45e+27], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision], N[(U + N[(l * N[(J * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -8.4000000000000007e215

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 57.3%

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

   3. Taylor expanded in K around 0 57.3%

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

   if -8.4000000000000007e215 < l < -5.99999999999999981e152

   1. Initial program 100.0%

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

   3. Applied egg-rr 64.5%

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

   if -5.99999999999999981e152 < l < -1100

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{-8 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 26.9%

         \[\leadsto \color{blue}{-8 - U \cdot U} \]

   7. Simplified 26.9%

      \[\leadsto \color{blue}{-8 - U \cdot U} \]

   if -1100 < l < 2.45000000000000007e27

   1. Initial program 73.5%

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

   2. Taylor expanded in l around 0 94.3%

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

   3. Taylor expanded in K around 0 85.8%

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

      1. +-commutative 85.8%

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

      2. associate-*r* 85.8%

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

      3. fma-def 85.8%

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

      4. *-commutative 85.8%

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

   5. Simplified 85.8%

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

   if 2.45000000000000007e27 < 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. Taylor expanded in l around 0 26.8%

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

   3. Taylor expanded in K around 0 34.1%

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

      1. associate-*r* 34.1%

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

      2. associate-*r* 34.1%

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

      3. associate-*r* 34.1%

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

      4. distribute-rgt-out 43.1%

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

      5. *-commutative 43.1%

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

      6. associate-*r* 43.1%

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

      7. distribute-rgt-out 43.1%

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

      8. unpow2 43.1%

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

   5. Simplified 43.1%

      \[\leadsto \color{blue}{U + \ell \cdot \left(J \cdot \left(-0.25 \cdot \left(K \cdot K\right) + 2\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.4 \cdot 10^{+215}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{+152}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1100:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 10: 56.2% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{+153}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1100:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* l J)))))
   (if (<= l -5.6e+215)
     t_0
     (if (<= l -4.4e+153)
       (* U U)
       (if (<= l -1100.0)
         (- -8.0 (* U U))
         (if (<= l 4.6e+28)
           t_0
           (+ U (* l (* J (+ 2.0 (* (* K K) -0.25)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * J));
	double tmp;
	if (l <= -5.6e+215) {
		tmp = t_0;
	} else if (l <= -4.4e+153) {
		tmp = U * U;
	} else if (l <= -1100.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 4.6e+28) {
		tmp = t_0;
	} else {
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (2.0d0 * (l * j))
    if (l <= (-5.6d+215)) then
        tmp = t_0
    else if (l <= (-4.4d+153)) then
        tmp = u * u
    else if (l <= (-1100.0d0)) then
        tmp = (-8.0d0) - (u * u)
    else if (l <= 4.6d+28) then
        tmp = t_0
    else
        tmp = u + (l * (j * (2.0d0 + ((k * k) * (-0.25d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * J));
	double tmp;
	if (l <= -5.6e+215) {
		tmp = t_0;
	} else if (l <= -4.4e+153) {
		tmp = U * U;
	} else if (l <= -1100.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 4.6e+28) {
		tmp = t_0;
	} else {
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (l * J))
	tmp = 0
	if l <= -5.6e+215:
		tmp = t_0
	elif l <= -4.4e+153:
		tmp = U * U
	elif l <= -1100.0:
		tmp = -8.0 - (U * U)
	elif l <= 4.6e+28:
		tmp = t_0
	else:
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(l * J)))
	tmp = 0.0
	if (l <= -5.6e+215)
		tmp = t_0;
	elseif (l <= -4.4e+153)
		tmp = Float64(U * U);
	elseif (l <= -1100.0)
		tmp = Float64(-8.0 - Float64(U * U));
	elseif (l <= 4.6e+28)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(Float64(K * K) * -0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (l * J));
	tmp = 0.0;
	if (l <= -5.6e+215)
		tmp = t_0;
	elseif (l <= -4.4e+153)
		tmp = U * U;
	elseif (l <= -1100.0)
		tmp = -8.0 - (U * U);
	elseif (l <= 4.6e+28)
		tmp = t_0;
	else
		tmp = U + (l * (J * (2.0 + ((K * K) * -0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.6e+215], t$95$0, If[LessEqual[l, -4.4e+153], N[(U * U), $MachinePrecision], If[LessEqual[l, -1100.0], N[(-8.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.6e+28], t$95$0, N[(U + N[(l * N[(J * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.5999999999999999e215 or -1100 < l < 4.59999999999999968e28

   1. Initial program 76.7%

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

   2. Taylor expanded in l around 0 89.9%

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

   3. Taylor expanded in K around 0 82.3%

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

   if -5.5999999999999999e215 < l < -4.3999999999999999e153

   1. Initial program 100.0%

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

   3. Applied egg-rr 64.5%

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

   if -4.3999999999999999e153 < l < -1100

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{-8 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 26.9%

         \[\leadsto \color{blue}{-8 - U \cdot U} \]

   7. Simplified 26.9%

      \[\leadsto \color{blue}{-8 - U \cdot U} \]

   if 4.59999999999999968e28 < 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. Taylor expanded in l around 0 26.8%

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

   3. Taylor expanded in K around 0 34.1%

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

      1. associate-*r* 34.1%

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

      2. associate-*r* 34.1%

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

      3. associate-*r* 34.1%

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

      4. distribute-rgt-out 43.1%

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

      5. *-commutative 43.1%

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

      6. associate-*r* 43.1%

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

      7. distribute-rgt-out 43.1%

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

      8. unpow2 43.1%

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

   5. Simplified 43.1%

      \[\leadsto \color{blue}{U + \ell \cdot \left(J \cdot \left(-0.25 \cdot \left(K \cdot K\right) + 2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+215}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{+153}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1100:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+28}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 11: 52.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{+155}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1200:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* l J)))))
   (if (<= l -5.6e+215)
     t_0
     (if (<= l -5.2e+155)
       (* U U)
       (if (<= l -1200.0)
         (- -8.0 (* U U))
         (if (<= l 3.8e+32) t_0 (+ U (* (* J (* K K)) (* l -0.25)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * J));
	double tmp;
	if (l <= -5.6e+215) {
		tmp = t_0;
	} else if (l <= -5.2e+155) {
		tmp = U * U;
	} else if (l <= -1200.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 3.8e+32) {
		tmp = t_0;
	} else {
		tmp = U + ((J * (K * K)) * (l * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (2.0d0 * (l * j))
    if (l <= (-5.6d+215)) then
        tmp = t_0
    else if (l <= (-5.2d+155)) then
        tmp = u * u
    else if (l <= (-1200.0d0)) then
        tmp = (-8.0d0) - (u * u)
    else if (l <= 3.8d+32) then
        tmp = t_0
    else
        tmp = u + ((j * (k * k)) * (l * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * J));
	double tmp;
	if (l <= -5.6e+215) {
		tmp = t_0;
	} else if (l <= -5.2e+155) {
		tmp = U * U;
	} else if (l <= -1200.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 3.8e+32) {
		tmp = t_0;
	} else {
		tmp = U + ((J * (K * K)) * (l * -0.25));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (l * J))
	tmp = 0
	if l <= -5.6e+215:
		tmp = t_0
	elif l <= -5.2e+155:
		tmp = U * U
	elif l <= -1200.0:
		tmp = -8.0 - (U * U)
	elif l <= 3.8e+32:
		tmp = t_0
	else:
		tmp = U + ((J * (K * K)) * (l * -0.25))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(l * J)))
	tmp = 0.0
	if (l <= -5.6e+215)
		tmp = t_0;
	elseif (l <= -5.2e+155)
		tmp = Float64(U * U);
	elseif (l <= -1200.0)
		tmp = Float64(-8.0 - Float64(U * U));
	elseif (l <= 3.8e+32)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(Float64(J * Float64(K * K)) * Float64(l * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (l * J));
	tmp = 0.0;
	if (l <= -5.6e+215)
		tmp = t_0;
	elseif (l <= -5.2e+155)
		tmp = U * U;
	elseif (l <= -1200.0)
		tmp = -8.0 - (U * U);
	elseif (l <= 3.8e+32)
		tmp = t_0;
	else
		tmp = U + ((J * (K * K)) * (l * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.6e+215], t$95$0, If[LessEqual[l, -5.2e+155], N[(U * U), $MachinePrecision], If[LessEqual[l, -1200.0], N[(-8.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e+32], t$95$0, N[(U + N[(N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(l * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.5999999999999999e215 or -1200 < l < 3.8000000000000003e32

   1. Initial program 76.9%

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

   2. Taylor expanded in l around 0 89.3%

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

   3. Taylor expanded in K around 0 81.8%

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

   if -5.5999999999999999e215 < l < -5.2000000000000004e155

   1. Initial program 100.0%

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

   3. Applied egg-rr 64.5%

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

   if -5.2000000000000004e155 < l < -1200

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{-8 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 26.9%

         \[\leadsto \color{blue}{-8 - U \cdot U} \]

   7. Simplified 26.9%

      \[\leadsto \color{blue}{-8 - U \cdot U} \]

   if 3.8000000000000003e32 < 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. Taylor expanded in l around 0 27.2%

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

   3. Taylor expanded in K around 0 34.6%

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

      1. associate-*r* 34.6%

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

      2. associate-*r* 34.6%

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

      3. associate-*r* 34.6%

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

      4. distribute-rgt-out 43.7%

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

      5. *-commutative 43.7%

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

      6. associate-*r* 43.7%

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

      7. distribute-rgt-out 43.7%

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

      8. unpow2 43.7%

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

   5. Simplified 43.7%

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

   6. Taylor expanded in K around inf 34.1%

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

      1. *-commutative 34.1%

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

      2. associate-*r* 34.1%

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

      3. associate-*l* 34.1%

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

      4. unpow2 34.1%

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

   8. Simplified 34.1%

      \[\leadsto U + \color{blue}{\left(J \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot -0.25\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+215}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{+155}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1200:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+32}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(K \cdot K\right)\right) \cdot \left(\ell \cdot -0.25\right)\\ \end{array} \]

Alternative 12: 43.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(U \cdot -8\right)\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+154}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1100:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+78}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (* U -8.0))))
   (if (<= l -3.6e+154)
     (* U U)
     (if (<= l -1000.0)
       t_0
       (if (<= l 1100.0) U (if (<= l 1.35e+78) (* U U) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (U * -8.0);
	double tmp;
	if (l <= -3.6e+154) {
		tmp = U * U;
	} else if (l <= -1000.0) {
		tmp = t_0;
	} else if (l <= 1100.0) {
		tmp = U;
	} else if (l <= 1.35e+78) {
		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 = u * (u * (-8.0d0))
    if (l <= (-3.6d+154)) then
        tmp = u * u
    else if (l <= (-1000.0d0)) then
        tmp = t_0
    else if (l <= 1100.0d0) then
        tmp = u
    else if (l <= 1.35d+78) 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 = U * (U * -8.0);
	double tmp;
	if (l <= -3.6e+154) {
		tmp = U * U;
	} else if (l <= -1000.0) {
		tmp = t_0;
	} else if (l <= 1100.0) {
		tmp = U;
	} else if (l <= 1.35e+78) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (U * -8.0)
	tmp = 0
	if l <= -3.6e+154:
		tmp = U * U
	elif l <= -1000.0:
		tmp = t_0
	elif l <= 1100.0:
		tmp = U
	elif l <= 1.35e+78:
		tmp = U * U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(U * -8.0))
	tmp = 0.0
	if (l <= -3.6e+154)
		tmp = Float64(U * U);
	elseif (l <= -1000.0)
		tmp = t_0;
	elseif (l <= 1100.0)
		tmp = U;
	elseif (l <= 1.35e+78)
		tmp = Float64(U * U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (U * -8.0);
	tmp = 0.0;
	if (l <= -3.6e+154)
		tmp = U * U;
	elseif (l <= -1000.0)
		tmp = t_0;
	elseif (l <= 1100.0)
		tmp = U;
	elseif (l <= 1.35e+78)
		tmp = U * U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(U * -8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.6e+154], N[(U * U), $MachinePrecision], If[LessEqual[l, -1000.0], t$95$0, If[LessEqual[l, 1100.0], U, If[LessEqual[l, 1.35e+78], N[(U * U), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.6000000000000001e154 or 1100 < l < 1.35000000000000002e78

   1. Initial program 100.0%

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

   3. Applied egg-rr 32.7%

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

   if -3.6000000000000001e154 < l < -1e3 or 1.35000000000000002e78 < 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. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in K around 0 71.8%

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

   7. Applied egg-rr 22.8%

      \[\leadsto \color{blue}{\left(U \cdot -8\right) \cdot U} \]

   if -1e3 < l < 1100

   1. Initial program 72.5%

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

   2. Taylor expanded in J around 0 70.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+154}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -1000:\\ \;\;\;\;U \cdot \left(U \cdot -8\right)\\ \mathbf{elif}\;\ell \leq 1100:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+78}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U \cdot -8\right)\\ \end{array} \]

Alternative 13: 43.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -8 - U \cdot U\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+155}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -480:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 2300:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+81}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- -8.0 (* U U))))
   (if (<= l -8.5e+155)
     (* U U)
     (if (<= l -480.0)
       t_0
       (if (<= l 2300.0) U (if (<= l 6.6e+81) (* U U) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = -8.0 - (U * U);
	double tmp;
	if (l <= -8.5e+155) {
		tmp = U * U;
	} else if (l <= -480.0) {
		tmp = t_0;
	} else if (l <= 2300.0) {
		tmp = U;
	} else if (l <= 6.6e+81) {
		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 = (-8.0d0) - (u * u)
    if (l <= (-8.5d+155)) then
        tmp = u * u
    else if (l <= (-480.0d0)) then
        tmp = t_0
    else if (l <= 2300.0d0) then
        tmp = u
    else if (l <= 6.6d+81) 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 = -8.0 - (U * U);
	double tmp;
	if (l <= -8.5e+155) {
		tmp = U * U;
	} else if (l <= -480.0) {
		tmp = t_0;
	} else if (l <= 2300.0) {
		tmp = U;
	} else if (l <= 6.6e+81) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = -8.0 - (U * U)
	tmp = 0
	if l <= -8.5e+155:
		tmp = U * U
	elif l <= -480.0:
		tmp = t_0
	elif l <= 2300.0:
		tmp = U
	elif l <= 6.6e+81:
		tmp = U * U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(-8.0 - Float64(U * U))
	tmp = 0.0
	if (l <= -8.5e+155)
		tmp = Float64(U * U);
	elseif (l <= -480.0)
		tmp = t_0;
	elseif (l <= 2300.0)
		tmp = U;
	elseif (l <= 6.6e+81)
		tmp = Float64(U * U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = -8.0 - (U * U);
	tmp = 0.0;
	if (l <= -8.5e+155)
		tmp = U * U;
	elseif (l <= -480.0)
		tmp = t_0;
	elseif (l <= 2300.0)
		tmp = U;
	elseif (l <= 6.6e+81)
		tmp = U * U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(-8.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8.5e+155], N[(U * U), $MachinePrecision], If[LessEqual[l, -480.0], t$95$0, If[LessEqual[l, 2300.0], U, If[LessEqual[l, 6.6e+81], N[(U * U), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.5000000000000002e155 or 2300 < l < 6.6e81

   1. Initial program 100.0%

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

   3. Applied egg-rr 32.0%

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

   if -8.5000000000000002e155 < l < -480 or 6.6e81 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{-8 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 23.2%

         \[\leadsto \color{blue}{-8 - U \cdot U} \]

   7. Simplified 23.2%

      \[\leadsto \color{blue}{-8 - U \cdot U} \]

   if -480 < l < 2300

   1. Initial program 72.5%

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

   2. Taylor expanded in J around 0 70.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+155}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -480:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 2300:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+81}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;-8 - U \cdot U\\ \end{array} \]

Alternative 14: 43.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+155}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -780:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 600:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+82}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.5e+155)
   (* U U)
   (if (<= l -780.0)
     (- -8.0 (* U U))
     (if (<= l 600.0) U (if (<= l 1.95e+82) (* U U) (* U (- 2.0 U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.5e+155) {
		tmp = U * U;
	} else if (l <= -780.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 600.0) {
		tmp = U;
	} else if (l <= 1.95e+82) {
		tmp = U * U;
	} else {
		tmp = U * (2.0 - U);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-2.5d+155)) then
        tmp = u * u
    else if (l <= (-780.0d0)) then
        tmp = (-8.0d0) - (u * u)
    else if (l <= 600.0d0) then
        tmp = u
    else if (l <= 1.95d+82) then
        tmp = u * u
    else
        tmp = u * (2.0d0 - u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.5e+155) {
		tmp = U * U;
	} else if (l <= -780.0) {
		tmp = -8.0 - (U * U);
	} else if (l <= 600.0) {
		tmp = U;
	} else if (l <= 1.95e+82) {
		tmp = U * U;
	} else {
		tmp = U * (2.0 - U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.5e+155:
		tmp = U * U
	elif l <= -780.0:
		tmp = -8.0 - (U * U)
	elif l <= 600.0:
		tmp = U
	elif l <= 1.95e+82:
		tmp = U * U
	else:
		tmp = U * (2.0 - U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.5e+155)
		tmp = Float64(U * U);
	elseif (l <= -780.0)
		tmp = Float64(-8.0 - Float64(U * U));
	elseif (l <= 600.0)
		tmp = U;
	elseif (l <= 1.95e+82)
		tmp = Float64(U * U);
	else
		tmp = Float64(U * Float64(2.0 - U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.5e+155)
		tmp = U * U;
	elseif (l <= -780.0)
		tmp = -8.0 - (U * U);
	elseif (l <= 600.0)
		tmp = U;
	elseif (l <= 1.95e+82)
		tmp = U * U;
	else
		tmp = U * (2.0 - U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.5e+155], N[(U * U), $MachinePrecision], If[LessEqual[l, -780.0], N[(-8.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 600.0], U, If[LessEqual[l, 1.95e+82], N[(U * U), $MachinePrecision], N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5e155 or 600 < l < 1.94999999999999988e82

   1. Initial program 100.0%

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

   3. Applied egg-rr 32.0%

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

   if -2.5e155 < l < -780

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{-8 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 26.9%

         \[\leadsto \color{blue}{-8 - U \cdot U} \]

   7. Simplified 26.9%

      \[\leadsto \color{blue}{-8 - U \cdot U} \]

   if -780 < l < 600

   1. Initial program 72.5%

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

   2. Taylor expanded in J around 0 70.2%

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

   if 1.94999999999999988e82 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 19.9%

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

      1. fma-udef 19.9%

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

      2. +-commutative 19.9%

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

      3. associate-+r+ 19.9%

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

      4. count-2 19.9%

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

      5. distribute-rgt-out 19.9%

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

   7. Simplified 19.9%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+155}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -780:\\ \;\;\;\;-8 - U \cdot U\\ \mathbf{elif}\;\ell \leq 600:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+82}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \end{array} \]

Alternative 15: 43.2% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -180000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 800:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -180000.0) (* U U) (if (<= l 800.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -180000.0) {
		tmp = U * U;
	} else if (l <= 800.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-180000.0d0)) then
        tmp = u * u
    else if (l <= 800.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -180000.0) {
		tmp = U * U;
	} else if (l <= 800.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -180000.0:
		tmp = U * U
	elif l <= 800.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -180000.0)
		tmp = Float64(U * U);
	elseif (l <= 800.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -180000.0)
		tmp = U * U;
	elseif (l <= 800.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -180000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 800.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8e5 or 800 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 17.8%

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

   if -1.8e5 < l < 800

   1. Initial program 72.9%

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

   2. Taylor expanded in J around 0 69.1%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -180000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 800:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 16: 54.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 + (2.0d0 * (l * j))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in l around 0 60.8%

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

3. Taylor expanded in K around 0 53.8%

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

4. Final simplification 53.8%

   \[\leadsto U + 2 \cdot \left(\ell \cdot J\right) \]

Alternative 17: 2.8% 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 86.5%

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

3. Applied egg-rr 2.9%

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

   1. *-inverses 2.9%

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

5. Simplified 2.9%

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

6. Final simplification 2.9%

   \[\leadsto 1 \]

Alternative 18: 37.4% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in J around 0 35.8%

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

3. Final simplification 35.8%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023278 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))