Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.5%

Time: 13.5s

Alternatives: 15

Speedup: 1.5×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.0)))
     (+ (* t_0 (* t_1 J)) U)
     (+
      U
      (* t_0 (+ (* 2.0 (* l J)) (* 0.3333333333333333 (* J (pow l 3.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.0)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * pow(l, 3.0)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.0)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * Math.pow(l, 3.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.0):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * math.pow(l, 3.0)))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.0)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * (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]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 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 -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 68.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 2: 86.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(U + N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 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 76.7%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 68.7%

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

   2. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 89.3%

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

Alternative 3: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;t_0 \leq -0.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;t_0 \leq 0.25:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))
   (if (<= t_0 -0.55)
     t_1
     (if (<= t_0 -0.01)
       (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
       (if (<= t_0 0.25)
         t_1
         (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (2.0 * (l * (J * cos((K * 0.5)))));
	double tmp;
	if (t_0 <= -0.55) {
		tmp = t_1;
	} else if (t_0 <= -0.01) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (t_0 <= 0.25) {
		tmp = t_1;
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    if (t_0 <= (-0.55d0)) then
        tmp = t_1
    else if (t_0 <= (-0.01d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else if (t_0 <= 0.25d0) then
        tmp = t_1
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	double tmp;
	if (t_0 <= -0.55) {
		tmp = t_1;
	} else if (t_0 <= -0.01) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (t_0 <= 0.25) {
		tmp = t_1;
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	tmp = 0
	if t_0 <= -0.55:
		tmp = t_1
	elif t_0 <= -0.01:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	elif t_0 <= 0.25:
		tmp = t_1
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))))
	tmp = 0.0
	if (t_0 <= -0.55)
		tmp = t_1;
	elseif (t_0 <= -0.01)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	elseif (t_0 <= 0.25)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (2.0 * (l * (J * cos((K * 0.5)))));
	tmp = 0.0;
	if (t_0 <= -0.55)
		tmp = t_1;
	elseif (t_0 <= -0.01)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	elseif (t_0 <= 0.25)
		tmp = t_1;
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.55], t$95$1, If[LessEqual[t$95$0, -0.01], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.25], t$95$1, N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.55000000000000004 or -0.0100000000000000002 < (cos.f64 (/.f64 K 2)) < 0.25

   1. Initial program 78.0%

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

   2. Taylor expanded in l around 0 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l* 80.2%

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

      3. *-commutative 80.2%

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

   4. Simplified 80.2%

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

   if -0.55000000000000004 < (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 87.2%

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

   2. Taylor expanded in l around 0 88.2%

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

   3. Taylor expanded in l around 0 46.6%

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

      1. *-commutative 46.6%

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

      2. *-commutative 46.6%

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

      3. associate-*r* 46.6%

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

      4. associate-*r* 46.6%

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

      5. *-commutative 46.6%

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

      6. associate-*l* 46.7%

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

      7. *-commutative 46.7%

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

      8. associate-*l* 46.7%

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

      9. *-commutative 46.7%

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

   5. Simplified 46.7%

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

   6. Taylor expanded in K around 0 51.4%

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

      1. associate-*r* 51.4%

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

      2. distribute-rgt-out 68.1%

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

      3. *-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. unpow2 68.1%

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

   8. Simplified 68.1%

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

   if 0.25 < (cos.f64 (/.f64 K 2))

   1. Initial program 83.3%

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

   2. Taylor expanded in l around 0 90.1%

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

      1. fma-def 90.1%

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

      2. +-commutative 90.1%

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

      3. associate-*r* 90.1%

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

      4. associate-*r* 90.1%

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

      5. distribute-rgt-out 90.1%

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

      6. *-commutative 90.1%

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

      7. *-commutative 90.1%

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

      8. div-inv 90.1%

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

      9. metadata-eval 90.1%

         \[\leadsto \mathsf{fma}\left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right), \cos \left(K \cdot \color{blue}{0.5}\right), U\right) \]

   4. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right), \cos \left(K \cdot 0.5\right), U\right)} \]

   5. Taylor expanded in K around 0 87.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.55:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.25:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 4: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.92:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* 0.3333333333333333 (* (pow l 3.0) (* J (cos (* K 0.5)))))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -6.6e+146)
     t_0
     (if (<= l -0.92)
       t_1
       (if (<= l 8.4e-16)
         (+
          U
          (*
           (cos (/ K 2.0))
           (+ (* 2.0 (* l J)) (* 0.3333333333333333 (* J (pow l 3.0))))))
         (if (<= l 6e+93) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (pow(l, 3.0) * (J * cos((K * 0.5)))));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -6.6e+146) {
		tmp = t_0;
	} else if (l <= -0.92) {
		tmp = t_1;
	} else if (l <= 8.4e-16) {
		tmp = U + (cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * pow(l, 3.0)))));
	} else if (l <= 6e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * ((l ** 3.0d0) * (j * cos((k * 0.5d0)))))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-6.6d+146)) then
        tmp = t_0
    else if (l <= (-0.92d0)) then
        tmp = t_1
    else if (l <= 8.4d-16) then
        tmp = u + (cos((k / 2.0d0)) * ((2.0d0 * (l * j)) + (0.3333333333333333d0 * (j * (l ** 3.0d0)))))
    else if (l <= 6d+93) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (Math.pow(l, 3.0) * (J * Math.cos((K * 0.5)))));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -6.6e+146) {
		tmp = t_0;
	} else if (l <= -0.92) {
		tmp = t_1;
	} else if (l <= 8.4e-16) {
		tmp = U + (Math.cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * Math.pow(l, 3.0)))));
	} else if (l <= 6e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (math.pow(l, 3.0) * (J * math.cos((K * 0.5)))))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -6.6e+146:
		tmp = t_0
	elif l <= -0.92:
		tmp = t_1
	elif l <= 8.4e-16:
		tmp = U + (math.cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * math.pow(l, 3.0)))))
	elif l <= 6e+93:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64((l ^ 3.0) * Float64(J * cos(Float64(K * 0.5))))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -6.6e+146)
		tmp = t_0;
	elseif (l <= -0.92)
		tmp = t_1;
	elseif (l <= 8.4e-16)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(Float64(2.0 * Float64(l * J)) + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))));
	elseif (l <= 6e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * ((l ^ 3.0) * (J * cos((K * 0.5)))));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -6.6e+146)
		tmp = t_0;
	elseif (l <= -0.92)
		tmp = t_1;
	elseif (l <= 8.4e-16)
		tmp = U + (cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * (l ^ 3.0)))));
	elseif (l <= 6e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.6e+146], t$95$0, If[LessEqual[l, -0.92], t$95$1, If[LessEqual[l, 8.4e-16], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+93], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.60000000000000032e146 or 5.99999999999999957e93 < 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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -6.60000000000000032e146 < l < -0.92000000000000004 or 8.4000000000000004e-16 < l < 5.99999999999999957e93

   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 92.9%

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

   if -0.92000000000000004 < l < 8.4000000000000004e-16

   1. Initial program 68.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+146}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.92:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+93}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.0305:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* 0.3333333333333333 (* (pow l 3.0) (* J (cos (* K 0.5)))))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -6.6e+146)
     t_0
     (if (<= l -0.0305)
       t_1
       (if (<= l 8.4e-16)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 1.95e+93) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (pow(l, 3.0) * (J * cos((K * 0.5)))));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -6.6e+146) {
		tmp = t_0;
	} else if (l <= -0.0305) {
		tmp = t_1;
	} else if (l <= 8.4e-16) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.95e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * ((l ** 3.0d0) * (j * cos((k * 0.5d0)))))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-6.6d+146)) then
        tmp = t_0
    else if (l <= (-0.0305d0)) then
        tmp = t_1
    else if (l <= 8.4d-16) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 1.95d+93) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (Math.pow(l, 3.0) * (J * Math.cos((K * 0.5)))));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -6.6e+146) {
		tmp = t_0;
	} else if (l <= -0.0305) {
		tmp = t_1;
	} else if (l <= 8.4e-16) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.95e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (math.pow(l, 3.0) * (J * math.cos((K * 0.5)))))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -6.6e+146:
		tmp = t_0
	elif l <= -0.0305:
		tmp = t_1
	elif l <= 8.4e-16:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 1.95e+93:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64((l ^ 3.0) * Float64(J * cos(Float64(K * 0.5))))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -6.6e+146)
		tmp = t_0;
	elseif (l <= -0.0305)
		tmp = t_1;
	elseif (l <= 8.4e-16)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 1.95e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * ((l ^ 3.0) * (J * cos((K * 0.5)))));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -6.6e+146)
		tmp = t_0;
	elseif (l <= -0.0305)
		tmp = t_1;
	elseif (l <= 8.4e-16)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 1.95e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.6e+146], t$95$0, If[LessEqual[l, -0.0305], t$95$1, If[LessEqual[l, 8.4e-16], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e+93], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.60000000000000032e146 or 1.9500000000000001e93 < 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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -6.60000000000000032e146 < l < -0.030499999999999999 or 8.4000000000000004e-16 < l < 1.9500000000000001e93

   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 92.9%

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

   if -0.030499999999999999 < l < 8.4000000000000004e-16

   1. Initial program 68.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+146}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0305:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 6: 93.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \cos \left(K \cdot 0.5\right)\\ t_1 := U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot t_0\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.00072:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, t_0, U\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (cos (* K 0.5))))
        (t_1 (+ U (* 0.3333333333333333 (* (pow l 3.0) t_0))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -5e+156)
     t_1
     (if (<= l -0.00072)
       t_2
       (if (<= l 8.4e-16) (fma (* l 2.0) t_0 U) (if (<= l 7e+93) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * cos((K * 0.5));
	double t_1 = U + (0.3333333333333333 * (pow(l, 3.0) * t_0));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -5e+156) {
		tmp = t_1;
	} else if (l <= -0.00072) {
		tmp = t_2;
	} else if (l <= 8.4e-16) {
		tmp = fma((l * 2.0), t_0, U);
	} else if (l <= 7e+93) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(J * cos(Float64(K * 0.5)))
	t_1 = Float64(U + Float64(0.3333333333333333 * Float64((l ^ 3.0) * t_0)))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -5e+156)
		tmp = t_1;
	elseif (l <= -0.00072)
		tmp = t_2;
	elseif (l <= 8.4e-16)
		tmp = fma(Float64(l * 2.0), t_0, U);
	elseif (l <= 7e+93)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.3333333333333333 * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+156], t$95$1, If[LessEqual[l, -0.00072], t$95$2, If[LessEqual[l, 8.4e-16], N[(N[(l * 2.0), $MachinePrecision] * t$95$0 + U), $MachinePrecision], If[LessEqual[l, 7e+93], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.99999999999999992e156 or 6.99999999999999996e93 < 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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4.99999999999999992e156 < l < -7.20000000000000045e-4 or 8.4000000000000004e-16 < l < 6.99999999999999996e93

   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 92.9%

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

   if -7.20000000000000045e-4 < l < 8.4000000000000004e-16

   1. Initial program 68.7%

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

   2. Taylor expanded in l around 0 99.9%

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

   3. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+156}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00072:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+93}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ t_1 := U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -480:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 460:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (log1p (expm1 (- (/ -8.0 U) U))))
        (t_1 (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))
   (if (<= l -8e+73)
     t_1
     (if (<= l -480.0)
       t_0
       (if (<= l 460.0)
         (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
         (if (<= l 6.8e+129) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = log1p(expm1(((-8.0 / U) - U)));
	double t_1 = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	double tmp;
	if (l <= -8e+73) {
		tmp = t_1;
	} else if (l <= -480.0) {
		tmp = t_0;
	} else if (l <= 460.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 6.8e+129) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.log1p(Math.expm1(((-8.0 / U) - U)));
	double t_1 = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	double tmp;
	if (l <= -8e+73) {
		tmp = t_1;
	} else if (l <= -480.0) {
		tmp = t_0;
	} else if (l <= 460.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 6.8e+129) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.log1p(math.expm1(((-8.0 / U) - U)))
	t_1 = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	tmp = 0
	if l <= -8e+73:
		tmp = t_1
	elif l <= -480.0:
		tmp = t_0
	elif l <= 460.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 6.8e+129:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = log1p(expm1(Float64(Float64(-8.0 / U) - U)))
	t_1 = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))))
	tmp = 0.0
	if (l <= -8e+73)
		tmp = t_1;
	elseif (l <= -480.0)
		tmp = t_0;
	elseif (l <= 460.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 6.8e+129)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Log[1 + N[(Exp[N[(N[(-8.0 / U), $MachinePrecision] - U), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8e+73], t$95$1, If[LessEqual[l, -480.0], t$95$0, If[LessEqual[l, 460.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.8e+129], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.99999999999999986e73 or 6.80000000000000036e129 < 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 96.5%

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

      1. fma-def 96.5%

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

      2. +-commutative 96.5%

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

      3. associate-*r* 96.5%

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

      4. associate-*r* 96.5%

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

      5. distribute-rgt-out 96.5%

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

      6. *-commutative 96.5%

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

      7. *-commutative 96.5%

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

      8. div-inv 96.5%

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

      9. metadata-eval 96.5%

         \[\leadsto \mathsf{fma}\left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right), \cos \left(K \cdot \color{blue}{0.5}\right), U\right) \]

   4. Applied egg-rr 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right), \cos \left(K \cdot 0.5\right), U\right)} \]

   5. Taylor expanded in K around 0 72.1%

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

   if -7.99999999999999986e73 < l < -480 or 460 < l < 6.80000000000000036e129

   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 3.6%

      \[\leadsto \color{blue}{\frac{-8}{U} - U} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 65.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)} \]

   5. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)} \]

   if -480 < l < 460

   1. Initial program 69.1%

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

   2. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+73}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -480:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 460:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00255 \lor \neg \left(\ell \leq 8.4 \cdot 10^{-16}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.00255) (not (<= l 8.4e-16)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (fma (* l 2.0) (* J (cos (* K 0.5))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00255) || !(l <= 8.4e-16)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = fma((l * 2.0), (J * cos((K * 0.5))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.00255) || !(l <= 8.4e-16))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = fma(Float64(l * 2.0), Float64(J * cos(Float64(K * 0.5))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.00255], N[Not[LessEqual[l, 8.4e-16]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0025500000000000002 or 8.4000000000000004e-16 < 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 76.7%

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

   if -0.0025500000000000002 < l < 8.4000000000000004e-16

   1. Initial program 68.7%

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

   2. Taylor expanded in l around 0 99.9%

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

   3. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00255 \lor \neg \left(\ell \leq 8.4 \cdot 10^{-16}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \]

Alternative 9: 57.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.01d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (l * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.01) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.01:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.01)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 83.9%

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

   2. Taylor expanded in l around 0 94.8%

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

   3. Taylor expanded in l around 0 62.7%

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

      1. *-commutative 62.7%

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

      2. *-commutative 62.7%

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

      3. associate-*r* 62.7%

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

      4. associate-*r* 62.7%

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

      5. *-commutative 62.7%

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

      6. associate-*l* 62.7%

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

      7. *-commutative 62.7%

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

      8. associate-*l* 62.7%

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

      9. *-commutative 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in K around 0 40.3%

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

      1. associate-*r* 40.3%

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

      2. distribute-rgt-out 51.2%

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

      3. *-commutative 51.2%

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

      4. *-commutative 51.2%

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

      5. unpow2 51.2%

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

   8. Simplified 51.2%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 82.6%

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

   2. Taylor expanded in l around 0 90.5%

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

   3. Taylor expanded in l around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-*r* 70.5%

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

      4. associate-*r* 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. *-commutative 70.5%

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

      8. associate-*l* 70.5%

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

      9. *-commutative 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in K around 0 66.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 10: 63.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* 2.0 (* l (* J (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	return U + (2.0 * (l * (J * cos((K * 0.5)))));
}

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 * cos((k * 0.5d0)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
}

Python

def code(J, l, K, U):
	return U + (2.0 * (l * (J * math.cos((K * 0.5)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
end

Wolfram

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

Derivation

1. Initial program 82.9%

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

2. Taylor expanded in l around 0 68.8%

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

   1. *-commutative 68.8%

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

   2. associate-*l* 68.9%

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

   3. *-commutative 68.9%

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

4. Simplified 68.9%

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

5. Final simplification 68.9%

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

Alternative 11: 42.0% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.45 \cdot 10^{+15}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 600000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.45e+15) (* U U) (if (<= l 600000000.0) U (* U (- U -8.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.45e+15) {
		tmp = U * U;
	} else if (l <= 600000000.0) {
		tmp = U;
	} else {
		tmp = U * (U - -8.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-4.45d+15)) then
        tmp = u * u
    else if (l <= 600000000.0d0) then
        tmp = u
    else
        tmp = u * (u - (-8.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.45e+15) {
		tmp = U * U;
	} else if (l <= 600000000.0) {
		tmp = U;
	} else {
		tmp = U * (U - -8.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.45e+15:
		tmp = U * U
	elif l <= 600000000.0:
		tmp = U
	else:
		tmp = U * (U - -8.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.45e+15)
		tmp = Float64(U * U);
	elseif (l <= 600000000.0)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4.45e+15)
		tmp = U * U;
	elseif (l <= 600000000.0)
		tmp = U;
	else
		tmp = U * (U - -8.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.45e+15], N[(U * U), $MachinePrecision], If[LessEqual[l, 600000000.0], U, N[(U * N[(U - -8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.45e15

   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 17.2%

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

   if -4.45e15 < l < 6e8

   1. Initial program 69.7%

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

   2. Taylor expanded in J around 0 67.1%

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

   if 6e8 < 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 16.9%

      \[\leadsto \color{blue}{U \cdot \left(U - -8\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.45 \cdot 10^{+15}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 600000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \end{array} \]

Alternative 12: 42.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 600000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.45e+16) (* U U) (if (<= l 600000000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.45e+16) {
		tmp = U * U;
	} else if (l <= 600000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.45d+16)) then
        tmp = u * u
    else if (l <= 600000000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.45e+16) {
		tmp = U * U;
	} else if (l <= 600000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.45e+16:
		tmp = U * U
	elif l <= 600000000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.45e+16)
		tmp = Float64(U * U);
	elseif (l <= 600000000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.45e+16)
		tmp = U * U;
	elseif (l <= 600000000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.45e+16], N[(U * U), $MachinePrecision], If[LessEqual[l, 600000000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.45e16 or 6e8 < 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 17.0%

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

   if -1.45e16 < l < 6e8

   1. Initial program 69.7%

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

   2. Taylor expanded in J around 0 67.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 600000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 13: 53.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

2. Taylor expanded in l around 0 91.4%

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

3. Taylor expanded in l around 0 68.8%

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

   1. *-commutative 68.8%

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

   2. *-commutative 68.8%

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

   3. associate-*r* 68.8%

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

   4. associate-*r* 68.8%

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

   5. *-commutative 68.8%

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

   6. associate-*l* 68.9%

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

   7. *-commutative 68.9%

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

   8. associate-*l* 68.9%

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

   9. *-commutative 68.9%

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

5. Simplified 68.9%

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

6. Taylor expanded in K around 0 59.5%

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

7. Final simplification 59.5%

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

Alternative 14: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Applied egg-rr 3.0%

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

   1. *-inverses 3.0%

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

5. Simplified 3.0%

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

6. Final simplification 3.0%

   \[\leadsto 1 \]

Alternative 15: 35.9% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

2. Taylor expanded in J around 0 38.9%

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

3. Final simplification 38.9%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023266 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))