Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 99.6%

Time: 12.2s

Alternatives: 19

Speedup: 1.4×

• 49.7% 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: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -0.1 \lor \neg \left(t_0 \leq 10^{-13}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.1) (not (<= t_0 1e-13)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+ U (* J (* (cos (* K 0.5)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.1) || !(t_0 <= 1e-13)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-0.1d0)) .or. (.not. (t_0 <= 1d-13))) then
        tmp = (cos((k / 2.0d0)) * (t_0 * j)) + u
    else
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -0.1) || !(t_0 <= 1e-13)) {
		tmp = (Math.cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -0.1) or not (t_0 <= 1e-13):
		tmp = (math.cos((K / 2.0)) * (t_0 * J)) + U
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.1) || !(t_0 <= 1e-13))
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)) + U);
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -0.1) || ~((t_0 <= 1e-13)))
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	else
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.1], N[Not[LessEqual[t$95$0, 1e-13]], $MachinePrecision]], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.10000000000000001 or 1e-13 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -0.10000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-13

   1. Initial program 69.5%

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

   2. Taylor expanded in l around 0 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. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.1 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 10^{-13}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_1 \leq -0.886:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;t_1 \leq -0.665:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;t_1 \leq -0.04:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.886)
     (+ U (* J (* t_0 (* l 2.0))))
     (if (<= t_1 -0.665)
       (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
       (if (<= t_1 -0.04)
         (+ U (* 2.0 (* l (* J t_0))))
         (+ U (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.886) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else if (t_1 <= -0.665) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (t_1 <= -0.04) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else {
		tmp = U + (J * ((pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = cos((k / 2.0d0))
    if (t_1 <= (-0.886d0)) then
        tmp = u + (j * (t_0 * (l * 2.0d0)))
    else if (t_1 <= (-0.665d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else if (t_1 <= (-0.04d0)) then
        tmp = u + (2.0d0 * (l * (j * t_0)))
    else
        tmp = u + (j * (((l ** 3.0d0) * 0.3333333333333333d0) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.886) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else if (t_1 <= -0.665) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (t_1 <= -0.04) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else {
		tmp = U + (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -0.886:
		tmp = U + (J * (t_0 * (l * 2.0)))
	elif t_1 <= -0.665:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	elif t_1 <= -0.04:
		tmp = U + (2.0 * (l * (J * t_0)))
	else:
		tmp = U + (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.886)
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(l * 2.0))));
	elseif (t_1 <= -0.665)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	elseif (t_1 <= -0.04)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * t_0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_1 <= -0.886)
		tmp = U + (J * (t_0 * (l * 2.0)));
	elseif (t_1 <= -0.665)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	elseif (t_1 <= -0.04)
		tmp = U + (2.0 * (l * (J * t_0)));
	else
		tmp = U + (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.886], N[(U + N[(J * N[(t$95$0 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.665], 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$1, -0.04], N[(U + N[(2.0 * N[(l * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.88600000000000001

   1. Initial program 85.9%

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

   2. Taylor expanded in l around 0 95.2%

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

   3. Taylor expanded in J around 0 95.2%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 95.3%

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

      2. *-commutative 95.3%

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

      3. fma-def 95.3%

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

      4. *-commutative 95.3%

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

      5. *-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. *-commutative 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in l around 0 76.3%

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

      1. *-commutative 76.3%

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

   8. Simplified 76.3%

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

   if -0.88600000000000001 < (cos.f64 (/.f64 K 2)) < -0.66500000000000004

   1. Initial program 92.1%

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

   2. Taylor expanded in l around 0 83.9%

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

   3. Taylor expanded in l around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. *-commutative 37.0%

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

      3. *-commutative 37.0%

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

      4. associate-*r* 37.0%

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

      5. *-commutative 37.0%

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

      6. associate-*r* 37.0%

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

      7. *-commutative 37.0%

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

   5. Simplified 37.0%

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

   6. Taylor expanded in K around 0 67.4%

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

      1. associate-*r* 67.4%

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

      2. distribute-rgt-out 75.8%

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

      3. *-commutative 75.8%

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

      4. unpow2 75.8%

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

   8. Simplified 75.8%

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

   if -0.66500000000000004 < (cos.f64 (/.f64 K 2)) < -0.0400000000000000008

   1. Initial program 78.9%

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

   2. Taylor expanded in l around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-*l* 74.4%

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

      3. *-commutative 74.4%

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

   4. Simplified 74.4%

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

   if -0.0400000000000000008 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.2%

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

   2. Taylor expanded in l around 0 89.6%

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

   3. Taylor expanded in K around 0 86.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.886:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.665:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \]

Alternative 3: 92.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+79}:\\ \;\;\;\;U + t_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+17} \lor \neg \left(\ell \leq 0.054\right) \land \ell \leq 1.46 \cdot 10^{+59}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -7e+79)
     (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
     (if (or (<= l -9e+17) (and (not (<= l 0.054)) (<= l 1.46e+59)))
       (+ U (* (- (exp l) (exp (- l))) J))
       (+ U (* t_0 (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= -7e+79) {
		tmp = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	} else if ((l <= -9e+17) || (!(l <= 0.054) && (l <= 1.46e+59))) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = U + (t_0 * (J * ((pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (l <= (-7d+79)) then
        tmp = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    else if ((l <= (-9d+17)) .or. (.not. (l <= 0.054d0)) .and. (l <= 1.46d+59)) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else
        tmp = u + (t_0 * (j * (((l ** 3.0d0) * 0.3333333333333333d0) + (l * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -7e+79) {
		tmp = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	} else if ((l <= -9e+17) || (!(l <= 0.054) && (l <= 1.46e+59))) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else {
		tmp = U + (t_0 * (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= -7e+79:
		tmp = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	elif (l <= -9e+17) or (not (l <= 0.054) and (l <= 1.46e+59)):
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	else:
		tmp = U + (t_0 * (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -7e+79)
		tmp = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))));
	elseif ((l <= -9e+17) || (!(l <= 0.054) && (l <= 1.46e+59)))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -7e+79)
		tmp = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	elseif ((l <= -9e+17) || (~((l <= 0.054)) && (l <= 1.46e+59)))
		tmp = U + ((exp(l) - exp(-l)) * J);
	else
		tmp = U + (t_0 * (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -7e+79], N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -9e+17], And[N[Not[LessEqual[l, 0.054]], $MachinePrecision], LessEqual[l, 1.46e+59]]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.99999999999999961e79

   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 96.8%

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

   3. Taylor expanded in l around inf 96.8%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*l* 96.8%

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

   5. Simplified 96.8%

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

   if -6.99999999999999961e79 < l < -9e17 or 0.0539999999999999994 < l < 1.45999999999999992e59

   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 87.5%

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

   if -9e17 < l < 0.0539999999999999994 or 1.45999999999999992e59 < l

   1. Initial program 78.7%

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

   2. Taylor expanded in l around 0 98.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+79}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+17} \lor \neg \left(\ell \leq 0.054\right) \land \ell \leq 1.46 \cdot 10^{+59}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.00023:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.46 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -7e+79)
     t_0
     (if (<= l -9e+17)
       t_1
       (if (<= l 0.00023)
         (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
         (if (<= l 1.46e+59) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -7e+79) {
		tmp = t_0;
	} else if (l <= -9e+17) {
		tmp = t_1;
	} else if (l <= 0.00023) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 1.46e+59) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-7d+79)) then
        tmp = t_0
    else if (l <= (-9d+17)) then
        tmp = t_1
    else if (l <= 0.00023d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (l <= 1.46d+59) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -7e+79) {
		tmp = t_0;
	} else if (l <= -9e+17) {
		tmp = t_1;
	} else if (l <= 0.00023) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 1.46e+59) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -7e+79:
		tmp = t_0
	elif l <= -9e+17:
		tmp = t_1
	elif l <= 0.00023:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 1.46e+59:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -7e+79)
		tmp = t_0;
	elseif (l <= -9e+17)
		tmp = t_1;
	elseif (l <= 0.00023)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 1.46e+59)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * ((l ^ 3.0) * (J * 0.3333333333333333)));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -7e+79)
		tmp = t_0;
	elseif (l <= -9e+17)
		tmp = t_1;
	elseif (l <= 0.00023)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (l <= 1.46e+59)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $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, -7e+79], t$95$0, If[LessEqual[l, -9e+17], t$95$1, If[LessEqual[l, 0.00023], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.46e+59], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.99999999999999961e79 or 1.45999999999999992e59 < 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%

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

   3. Taylor expanded in l around inf 96.5%

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

      1. associate-*r* 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*l* 96.5%

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

   5. Simplified 96.5%

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

   if -6.99999999999999961e79 < l < -9e17 or 2.3000000000000001e-4 < l < 1.45999999999999992e59

   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 87.5%

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

   if -9e17 < l < 2.3000000000000001e-4

   1. Initial program 70.2%

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

   2. Taylor expanded in l around 0 98.8%

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

   3. Taylor expanded in J around 0 98.8%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 98.8%

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

      2. *-commutative 98.8%

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

      3. fma-def 98.8%

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

      4. *-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. *-commutative 98.8%

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

      7. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in l around 0 98.6%

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

      1. *-commutative 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+79}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+17}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.00023:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.46 \cdot 10^{+59}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 5: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ t_1 := U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -520:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (log1p (expm1 (- (/ -8.0 U) U))))
        (t_1 (+ U (* J (* (pow l 3.0) 0.3333333333333333)))))
   (if (<= l -3.9e+102)
     t_1
     (if (<= l -520.0)
       t_0
       (if (<= l 6200000000.0)
         (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
         (if (<= l 1.9e+99) 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 * (pow(l, 3.0) * 0.3333333333333333));
	double tmp;
	if (l <= -3.9e+102) {
		tmp = t_1;
	} else if (l <= -520.0) {
		tmp = t_0;
	} else if (l <= 6200000000.0) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 1.9e+99) {
		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 * (Math.pow(l, 3.0) * 0.3333333333333333));
	double tmp;
	if (l <= -3.9e+102) {
		tmp = t_1;
	} else if (l <= -520.0) {
		tmp = t_0;
	} else if (l <= 6200000000.0) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 1.9e+99) {
		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 * (math.pow(l, 3.0) * 0.3333333333333333))
	tmp = 0
	if l <= -3.9e+102:
		tmp = t_1
	elif l <= -520.0:
		tmp = t_0
	elif l <= 6200000000.0:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 1.9e+99:
		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((l ^ 3.0) * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -3.9e+102)
		tmp = t_1;
	elseif (l <= -520.0)
		tmp = t_0;
	elseif (l <= 6200000000.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 1.9e+99)
		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[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.9e+102], t$95$1, If[LessEqual[l, -520.0], t$95$0, If[LessEqual[l, 6200000000.0], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.9e+99], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.8999999999999998e102 or 1.9e99 < 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%

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

   3. Taylor expanded in K around 0 79.8%

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

   4. Taylor expanded in l around inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. *-commutative 79.8%

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

   6. Simplified 79.8%

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

   if -3.8999999999999998e102 < l < -520 or 6.2e9 < l < 1.9e99

   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 3.5%

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

      1. log1p-expm1-u 59.4%

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

   5. Applied egg-rr 59.4%

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

   if -520 < l < 6.2e9

   1. Initial program 70.7%

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

   2. Taylor expanded in l around 0 97.3%

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

   3. Taylor expanded in J around 0 97.3%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 97.3%

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

      2. *-commutative 97.3%

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

      3. fma-def 97.3%

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

      4. *-commutative 97.3%

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

      5. *-commutative 97.3%

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

      6. *-commutative 97.3%

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

      7. *-commutative 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in l around 0 97.2%

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

      1. *-commutative 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+102}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -520:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+17} \lor \neg \left(\ell \leq 0.00032\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -9e+17) (not (<= l 0.00032)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -9e+17) || !(l <= 0.00032)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-9d+17)) .or. (.not. (l <= 0.00032d0))) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -9e+17) || !(l <= 0.00032)) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -9e+17) or not (l <= 0.00032):
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -9e+17) || !(l <= 0.00032))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -9e+17) || ~((l <= 0.00032)))
		tmp = U + ((exp(l) - exp(-l)) * J);
	else
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -9e+17], N[Not[LessEqual[l, 0.00032]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9e17 or 3.20000000000000026e-4 < 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 78.9%

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

   if -9e17 < l < 3.20000000000000026e-4

   1. Initial program 70.2%

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

   2. Taylor expanded in l around 0 98.8%

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

   3. Taylor expanded in J around 0 98.8%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 98.8%

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

      2. *-commutative 98.8%

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

      3. fma-def 98.8%

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

      4. *-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. *-commutative 98.8%

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

      7. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in l around 0 98.6%

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

      1. *-commutative 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+17} \lor \neg \left(\ell \leq 0.00032\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 7: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot 0.3333333333333333\\ \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+90}:\\ \;\;\;\;J \cdot \left(t_0 + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) 0.3333333333333333)))
   (if (<= l -2.05e+90)
     (* J (+ t_0 (* l 2.0)))
     (if (<= l 6200000000.0)
       (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
       (if (<= l 6.5e+95)
         (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))
         (+ U (* J t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * 0.3333333333333333;
	double tmp;
	if (l <= -2.05e+90) {
		tmp = J * (t_0 + (l * 2.0));
	} else if (l <= 6200000000.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 6.5e+95) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = U + (J * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l ** 3.0d0) * 0.3333333333333333d0
    if (l <= (-2.05d+90)) then
        tmp = j * (t_0 + (l * 2.0d0))
    else if (l <= 6200000000.0d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else if (l <= 6.5d+95) then
        tmp = ((64.0d0 / (u * u)) - (u * u)) / (u + ((-8.0d0) / u))
    else
        tmp = u + (j * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * 0.3333333333333333;
	double tmp;
	if (l <= -2.05e+90) {
		tmp = J * (t_0 + (l * 2.0));
	} else if (l <= 6200000000.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 6.5e+95) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = U + (J * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * 0.3333333333333333
	tmp = 0
	if l <= -2.05e+90:
		tmp = J * (t_0 + (l * 2.0))
	elif l <= 6200000000.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 6.5e+95:
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U))
	else:
		tmp = U + (J * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * 0.3333333333333333)
	tmp = 0.0
	if (l <= -2.05e+90)
		tmp = Float64(J * Float64(t_0 + Float64(l * 2.0)));
	elseif (l <= 6200000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 6.5e+95)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)));
	else
		tmp = Float64(U + Float64(J * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * 0.3333333333333333;
	tmp = 0.0;
	if (l <= -2.05e+90)
		tmp = J * (t_0 + (l * 2.0));
	elseif (l <= 6200000000.0)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	elseif (l <= 6.5e+95)
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	else
		tmp = U + (J * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[l, -2.05e+90], N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6200000000.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+95], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.05000000000000021e90

   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 98.3%

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

   3. Taylor expanded in K around 0 79.0%

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

   4. Taylor expanded in J around inf 79.0%

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

   if -2.05000000000000021e90 < l < 6.2e9

   1. Initial program 74.4%

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

   2. Taylor expanded in l around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l* 88.0%

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

      3. *-commutative 88.0%

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

   4. Simplified 88.0%

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

   if 6.2e9 < l < 6.5e95

   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 4.3%

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

      1. flip-- 54.7%

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

      2. div-inv 54.7%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 54.7%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 54.7%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      5. +-commutative 54.7%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\color{blue}{U + \frac{-8}{U}}} \]

   5. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U + \frac{-8}{U}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 54.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot 1}{U + \frac{-8}{U}}} \]

      2. *-rgt-identity 54.7%

         \[\leadsto \frac{\color{blue}{\frac{64}{U \cdot U} - U \cdot U}}{U + \frac{-8}{U}} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}} \]

   if 6.5e95 < 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%

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

   3. Taylor expanded in K around 0 79.1%

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

   4. Taylor expanded in l around inf 79.1%

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

      1. associate-*r* 79.1%

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

      2. *-commutative 79.1%

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

   6. Simplified 79.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+90}:\\ \;\;\;\;J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 8: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot 0.3333333333333333\\ \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+90}:\\ \;\;\;\;J \cdot \left(t_0 + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) 0.3333333333333333)))
   (if (<= l -2.05e+90)
     (* J (+ t_0 (* l 2.0)))
     (if (<= l 6200000000.0)
       (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
       (if (<= l 5.2e+97)
         (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))
         (+ U (* J t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * 0.3333333333333333;
	double tmp;
	if (l <= -2.05e+90) {
		tmp = J * (t_0 + (l * 2.0));
	} else if (l <= 6200000000.0) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 5.2e+97) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = U + (J * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l ** 3.0d0) * 0.3333333333333333d0
    if (l <= (-2.05d+90)) then
        tmp = j * (t_0 + (l * 2.0d0))
    else if (l <= 6200000000.0d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (l <= 5.2d+97) then
        tmp = ((64.0d0 / (u * u)) - (u * u)) / (u + ((-8.0d0) / u))
    else
        tmp = u + (j * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * 0.3333333333333333;
	double tmp;
	if (l <= -2.05e+90) {
		tmp = J * (t_0 + (l * 2.0));
	} else if (l <= 6200000000.0) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 5.2e+97) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = U + (J * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * 0.3333333333333333
	tmp = 0
	if l <= -2.05e+90:
		tmp = J * (t_0 + (l * 2.0))
	elif l <= 6200000000.0:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 5.2e+97:
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U))
	else:
		tmp = U + (J * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * 0.3333333333333333)
	tmp = 0.0
	if (l <= -2.05e+90)
		tmp = Float64(J * Float64(t_0 + Float64(l * 2.0)));
	elseif (l <= 6200000000.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 5.2e+97)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)));
	else
		tmp = Float64(U + Float64(J * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * 0.3333333333333333;
	tmp = 0.0;
	if (l <= -2.05e+90)
		tmp = J * (t_0 + (l * 2.0));
	elseif (l <= 6200000000.0)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (l <= 5.2e+97)
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	else
		tmp = U + (J * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[l, -2.05e+90], N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6200000000.0], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e+97], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.05000000000000021e90

   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 98.3%

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

   3. Taylor expanded in K around 0 79.0%

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

   4. Taylor expanded in J around inf 79.0%

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

   if -2.05000000000000021e90 < l < 6.2e9

   1. Initial program 74.4%

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

   2. Taylor expanded in l around 0 88.9%

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

   3. Taylor expanded in J around 0 88.9%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 88.9%

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

      2. *-commutative 88.9%

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

      3. fma-def 88.9%

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

      4. *-commutative 88.9%

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

      5. *-commutative 88.9%

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

      6. *-commutative 88.9%

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

      7. *-commutative 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in l around 0 88.1%

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

      1. *-commutative 88.1%

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

   8. Simplified 88.1%

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

   if 6.2e9 < l < 5.2e97

   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 4.3%

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

      1. flip-- 54.7%

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

      2. div-inv 54.7%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 54.7%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 54.7%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      5. +-commutative 54.7%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\color{blue}{U + \frac{-8}{U}}} \]

   5. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U + \frac{-8}{U}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 54.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot 1}{U + \frac{-8}{U}}} \]

      2. *-rgt-identity 54.7%

         \[\leadsto \frac{\color{blue}{\frac{64}{U \cdot U} - U \cdot U}}{U + \frac{-8}{U}} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}} \]

   if 5.2e97 < 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%

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

   3. Taylor expanded in K around 0 79.1%

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

   4. Taylor expanded in l around inf 79.1%

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

      1. associate-*r* 79.1%

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

      2. *-commutative 79.1%

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

   6. Simplified 79.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+90}:\\ \;\;\;\;J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 9: 71.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* (pow l 3.0) 0.3333333333333333)))))
   (if (<= l -9e+17)
     t_0
     (if (<= l 6200000000.0)
       (fma l (* J 2.0) U)
       (if (<= l 2e+96)
         (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (pow(l, 3.0) * 0.3333333333333333));
	double tmp;
	if (l <= -9e+17) {
		tmp = t_0;
	} else if (l <= 6200000000.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 2e+96) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64((l ^ 3.0) * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -9e+17)
		tmp = t_0;
	elseif (l <= 6200000000.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 2e+96)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9e+17], t$95$0, If[LessEqual[l, 6200000000.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 2e+96], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9e17 or 2.0000000000000001e96 < 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 89.3%

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

   3. Taylor expanded in K around 0 70.5%

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

   4. Taylor expanded in l around inf 70.5%

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

      1. associate-*r* 70.5%

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

      2. *-commutative 70.5%

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

   6. Simplified 70.5%

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

   if -9e17 < l < 6.2e9

   1. Initial program 70.9%

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

   2. Taylor expanded in l around 0 96.5%

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

   3. Taylor expanded in l around 0 96.4%

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

      1. *-commutative 96.4%

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

      2. *-commutative 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*r* 96.4%

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

      5. *-commutative 96.4%

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

      6. associate-*r* 96.4%

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

      7. *-commutative 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in K around 0 83.7%

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

      1. associate-*r* 83.7%

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

      2. *-commutative 83.7%

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

      3. associate-*l* 83.7%

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

      4. fma-def 83.7%

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

   8. Simplified 83.7%

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

   if 6.2e9 < l < 2.0000000000000001e96

   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 4.3%

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

      1. flip-- 54.7%

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

      2. div-inv 54.7%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 54.7%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 54.7%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      5. +-commutative 54.7%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\color{blue}{U + \frac{-8}{U}}} \]

   5. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U + \frac{-8}{U}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 54.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot 1}{U + \frac{-8}{U}}} \]

      2. *-rgt-identity 54.7%

         \[\leadsto \frac{\color{blue}{\frac{64}{U \cdot U} - U \cdot U}}{U + \frac{-8}{U}} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+17}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 10: 71.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot 0.3333333333333333\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;J \cdot \left(t_0 + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) 0.3333333333333333)))
   (if (<= l -1.4e+18)
     (* J (+ t_0 (* l 2.0)))
     (if (<= l 6200000000.0)
       (fma l (* J 2.0) U)
       (if (<= l 1.85e+96)
         (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))
         (+ U (* J t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * 0.3333333333333333;
	double tmp;
	if (l <= -1.4e+18) {
		tmp = J * (t_0 + (l * 2.0));
	} else if (l <= 6200000000.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 1.85e+96) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = U + (J * t_0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * 0.3333333333333333)
	tmp = 0.0
	if (l <= -1.4e+18)
		tmp = Float64(J * Float64(t_0 + Float64(l * 2.0)));
	elseif (l <= 6200000000.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 1.85e+96)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)));
	else
		tmp = Float64(U + Float64(J * t_0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[l, -1.4e+18], N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6200000000.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.85e+96], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4e18

   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 83.0%

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

   3. Taylor expanded in K around 0 65.5%

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

   4. Taylor expanded in J around inf 65.5%

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

   if -1.4e18 < l < 6.2e9

   1. Initial program 70.9%

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

   2. Taylor expanded in l around 0 96.5%

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

   3. Taylor expanded in l around 0 96.4%

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

      1. *-commutative 96.4%

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

      2. *-commutative 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*r* 96.4%

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

      5. *-commutative 96.4%

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

      6. associate-*r* 96.4%

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

      7. *-commutative 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in K around 0 83.7%

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

      1. associate-*r* 83.7%

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

      2. *-commutative 83.7%

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

      3. associate-*l* 83.7%

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

      4. fma-def 83.7%

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

   8. Simplified 83.7%

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

   if 6.2e9 < l < 1.84999999999999996e96

   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 4.3%

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

      1. flip-- 54.7%

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

      2. div-inv 54.7%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 54.7%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 54.7%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      5. +-commutative 54.7%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\color{blue}{U + \frac{-8}{U}}} \]

   5. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U + \frac{-8}{U}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 54.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot 1}{U + \frac{-8}{U}}} \]

      2. *-rgt-identity 54.7%

         \[\leadsto \frac{\color{blue}{\frac{64}{U \cdot U} - U \cdot U}}{U + \frac{-8}{U}} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}} \]

   if 1.84999999999999996e96 < 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%

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

   3. Taylor expanded in K around 0 79.1%

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

   4. Taylor expanded in l around inf 79.1%

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

      1. associate-*r* 79.1%

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

      2. *-commutative 79.1%

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

   6. Simplified 79.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 11: 56.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -5.4e+251)
     t_0
     (if (<= l 6200000000.0)
       (fma l (* J 2.0) U)
       (if (<= l 9e+169)
         (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -5.4e+251) {
		tmp = t_0;
	} else if (l <= 6200000000.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 9e+169) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -5.4e+251)
		tmp = t_0;
	elseif (l <= 6200000000.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 9e+169)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.4e+251], t$95$0, If[LessEqual[l, 6200000000.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 9e+169], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.4000000000000002e251 or 8.9999999999999999e169 < 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%

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

   3. Taylor expanded in l around 0 36.1%

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

      1. *-commutative 36.1%

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

      2. *-commutative 36.1%

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

      3. *-commutative 36.1%

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

      4. associate-*r* 36.1%

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

      5. *-commutative 36.1%

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

      6. associate-*r* 36.1%

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

      7. *-commutative 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in K around 0 23.7%

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

      1. associate-*r* 23.7%

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

      2. distribute-rgt-out 53.7%

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

      3. *-commutative 53.7%

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

      4. unpow2 53.7%

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

   8. Simplified 53.7%

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

   if -5.4000000000000002e251 < l < 6.2e9

   1. Initial program 80.5%

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

   2. Taylor expanded in l around 0 91.0%

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

   3. Taylor expanded in l around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. *-commutative 76.3%

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

      3. *-commutative 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. associate-*r* 76.3%

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

      7. *-commutative 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in K around 0 67.2%

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

      1. associate-*r* 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 67.2%

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

      4. fma-def 67.2%

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

   8. Simplified 67.2%

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

   if 6.2e9 < l < 8.9999999999999999e169

   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 3.7%

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

      1. flip-- 40.6%

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

      2. div-inv 40.6%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 40.6%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 40.6%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      5. +-commutative 40.6%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\color{blue}{U + \frac{-8}{U}}} \]

   5. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U + \frac{-8}{U}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 40.6%

         \[\leadsto \color{blue}{\frac{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot 1}{U + \frac{-8}{U}}} \]

      2. *-rgt-identity 40.6%

         \[\leadsto \frac{\color{blue}{\frac{64}{U \cdot U} - U \cdot U}}{U + \frac{-8}{U}} \]

   7. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+251}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 12: 56.9% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -5.9e+251)
     t_0
     (if (<= l 6200000000.0)
       (+ U (* l (* J 2.0)))
       (if (<= l 8.5e+169)
         (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -5.9e+251) {
		tmp = t_0;
	} else if (l <= 6200000000.0) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= 8.5e+169) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / 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 + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    if (l <= (-5.9d+251)) then
        tmp = t_0
    else if (l <= 6200000000.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else if (l <= 8.5d+169) then
        tmp = ((64.0d0 / (u * u)) - (u * u)) / (u + ((-8.0d0) / 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 + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -5.9e+251) {
		tmp = t_0;
	} else if (l <= 6200000000.0) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= 8.5e+169) {
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	tmp = 0
	if l <= -5.9e+251:
		tmp = t_0
	elif l <= 6200000000.0:
		tmp = U + (l * (J * 2.0))
	elif l <= 8.5e+169:
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -5.9e+251)
		tmp = t_0;
	elseif (l <= 6200000000.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif (l <= 8.5e+169)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	tmp = 0.0;
	if (l <= -5.9e+251)
		tmp = t_0;
	elseif (l <= 6200000000.0)
		tmp = U + (l * (J * 2.0));
	elseif (l <= 8.5e+169)
		tmp = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.9e+251], t$95$0, If[LessEqual[l, 6200000000.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e+169], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.8999999999999999e251 or 8.5000000000000004e169 < 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%

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

   3. Taylor expanded in l around 0 36.1%

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

      1. *-commutative 36.1%

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

      2. *-commutative 36.1%

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

      3. *-commutative 36.1%

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

      4. associate-*r* 36.1%

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

      5. *-commutative 36.1%

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

      6. associate-*r* 36.1%

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

      7. *-commutative 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in K around 0 23.7%

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

      1. associate-*r* 23.7%

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

      2. distribute-rgt-out 53.7%

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

      3. *-commutative 53.7%

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

      4. unpow2 53.7%

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

   8. Simplified 53.7%

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

   if -5.8999999999999999e251 < l < 6.2e9

   1. Initial program 80.5%

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

   2. Taylor expanded in l around 0 91.0%

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

   3. Taylor expanded in l around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. *-commutative 76.3%

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

      3. *-commutative 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. associate-*r* 76.3%

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

      7. *-commutative 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in K around 0 67.2%

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

      1. associate-*r* 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 67.2%

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

   8. Simplified 67.2%

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

   if 6.2e9 < l < 8.5000000000000004e169

   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 3.7%

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

      1. flip-- 40.6%

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

      2. div-inv 40.6%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 40.6%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 40.6%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      5. +-commutative 40.6%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\color{blue}{U + \frac{-8}{U}}} \]

   5. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U + \frac{-8}{U}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 40.6%

         \[\leadsto \color{blue}{\frac{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot 1}{U + \frac{-8}{U}}} \]

      2. *-rgt-identity 40.6%

         \[\leadsto \frac{\color{blue}{\frac{64}{U \cdot U} - U \cdot U}}{U + \frac{-8}{U}} \]

   7. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+251}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 6200000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 13: 57.3% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+251} \lor \neg \left(\ell \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.05e+251) (not (<= l 3e+14)))
   (+ 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 ((l <= -2.05e+251) || !(l <= 3e+14)) {
		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 ((l <= (-2.05d+251)) .or. (.not. (l <= 3d+14))) 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 ((l <= -2.05e+251) || !(l <= 3e+14)) {
		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 (l <= -2.05e+251) or not (l <= 3e+14):
		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 ((l <= -2.05e+251) || !(l <= 3e+14))
		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 ((l <= -2.05e+251) || ~((l <= 3e+14)))
		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[Or[LessEqual[l, -2.05e+251], N[Not[LessEqual[l, 3e+14]], $MachinePrecision]], 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 l < -2.0500000000000001e251 or 3e14 < 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 88.7%

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

   3. Taylor expanded in l around 0 26.0%

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

      1. *-commutative 26.0%

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

      2. *-commutative 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r* 26.0%

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

      5. *-commutative 26.0%

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

      6. associate-*r* 26.0%

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

      7. *-commutative 26.0%

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

   5. Simplified 26.0%

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

   6. Taylor expanded in K around 0 21.2%

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

      1. associate-*r* 21.2%

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

      2. distribute-rgt-out 42.1%

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

      3. *-commutative 42.1%

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

      4. unpow2 42.1%

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

   8. Simplified 42.1%

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

   if -2.0500000000000001e251 < l < 3e14

   1. Initial program 80.6%

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

   2. Taylor expanded in l around 0 90.5%

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

   3. Taylor expanded in l around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-*r* 75.9%

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

      5. *-commutative 75.9%

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

      6. associate-*r* 75.9%

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

      7. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in K around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. *-commutative 66.8%

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

      3. associate-*l* 66.8%

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

   8. Simplified 66.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+251} \lor \neg \left(\ell \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 14: 53.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 8.2 \cdot 10^{+238}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(512 + \left(K \cdot K\right) \cdot -64\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 8.2e+238)
   (+ U (* l (* J 2.0)))
   (+ U (* J (+ 512.0 (* (* K K) -64.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 8.2e+238) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (J * (512.0 + ((K * K) * -64.0)));
	}
	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 (k <= 8.2d+238) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u + (j * (512.0d0 + ((k * k) * (-64.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 8.2e+238) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (J * (512.0 + ((K * K) * -64.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 8.2e+238:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U + (J * (512.0 + ((K * K) * -64.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 8.2e+238)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U + Float64(J * Float64(512.0 + Float64(Float64(K * K) * -64.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 8.2e+238)
		tmp = U + (l * (J * 2.0));
	else
		tmp = U + (J * (512.0 + ((K * K) * -64.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 8.2e+238], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(512.0 + N[(N[(K * K), $MachinePrecision] * -64.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 8.1999999999999998e238

   1. Initial program 85.8%

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

   2. Taylor expanded in l around 0 90.4%

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

   3. Taylor expanded in l around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. *-commutative 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-*r* 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*r* 63.4%

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

      7. *-commutative 63.4%

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

   5. Simplified 63.4%

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

   6. Taylor expanded in K around 0 54.6%

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

      1. associate-*r* 54.6%

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

      2. *-commutative 54.6%

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

      3. associate-*l* 54.6%

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

   8. Simplified 54.6%

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

   if 8.1999999999999998e238 < K

   1. Initial program 83.9%

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

   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 42.1%

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

      1. +-commutative 42.1%

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

      2. associate-*r* 42.1%

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

      3. distribute-rgt-out 42.1%

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

      4. *-commutative 42.1%

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

      5. unpow2 42.1%

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

   6. Simplified 42.1%

      \[\leadsto \color{blue}{J \cdot \left(512 + \left(K \cdot K\right) \cdot -64\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 8.2 \cdot 10^{+238}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(512 + \left(K \cdot K\right) \cdot -64\right)\\ \end{array} \]

Alternative 15: 42.2% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+68}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 850:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.8e+68) (* U U) (if (<= l 850.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.8e+68) {
		tmp = U * U;
	} else if (l <= 850.0) {
		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 <= (-4.8d+68)) then
        tmp = u * u
    else if (l <= 850.0d0) 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 <= -4.8e+68) {
		tmp = U * U;
	} else if (l <= 850.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.8e+68:
		tmp = U * U
	elif l <= 850.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.8e+68)
		tmp = Float64(U * U);
	elseif (l <= 850.0)
		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 <= -4.8e+68)
		tmp = U * U;
	elseif (l <= 850.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.8e+68], N[(U * U), $MachinePrecision], If[LessEqual[l, 850.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.80000000000000016e68 or 850 < 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 14.9%

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

   if -4.80000000000000016e68 < l < 850

   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 62.3%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+68}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 850:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 16: 42.2% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+68}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.8e+68) (* U (- U -8.0)) (if (<= l 1050.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.8e+68) {
		tmp = U * (U - -8.0);
	} else if (l <= 1050.0) {
		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 <= (-4.8d+68)) then
        tmp = u * (u - (-8.0d0))
    else if (l <= 1050.0d0) 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 <= -4.8e+68) {
		tmp = U * (U - -8.0);
	} else if (l <= 1050.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.8e+68:
		tmp = U * (U - -8.0)
	elif l <= 1050.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.8e+68)
		tmp = Float64(U * Float64(U - -8.0));
	elseif (l <= 1050.0)
		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 <= -4.8e+68)
		tmp = U * (U - -8.0);
	elseif (l <= 1050.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.8e+68], N[(U * N[(U - -8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1050.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.80000000000000016e68

   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 14.6%

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

   if -4.80000000000000016e68 < l < 1050

   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 62.3%

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

   if 1050 < 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 15.2%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+68}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 17: 54.2% 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 85.7%

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

2. Taylor expanded in l around 0 90.1%

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

3. Taylor expanded in l around 0 62.9%

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

   1. *-commutative 62.9%

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

   2. *-commutative 62.9%

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

   3. *-commutative 62.9%

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

   4. associate-*r* 62.9%

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

   5. *-commutative 62.9%

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

   6. associate-*r* 62.9%

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

   7. *-commutative 62.9%

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

5. Simplified 62.9%

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

6. Taylor expanded in K around 0 52.9%

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

   1. associate-*r* 52.9%

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

   2. *-commutative 52.9%

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

   3. associate-*l* 52.9%

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

8. Simplified 52.9%

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

9. Final simplification 52.9%

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

Alternative 18: 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 85.7%

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

3. Applied egg-rr 2.7%

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

   1. *-inverses 2.7%

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

5. Simplified 2.7%

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

6. Final simplification 2.7%

   \[\leadsto 1 \]

Alternative 19: 36.8% 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 85.7%

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

2. Taylor expanded in J around 0 34.0%

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

3. Final simplification 34.0%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023224 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))