Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.8%

Time: 16.4s

Alternatives: 14

Speedup: 1.4×

• 49.9% 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: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+24} \lor \neg \left(t_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -2e+24) (not (<= t_0 2e-6)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+
      U
      (*
       J
       (* (fma 0.3333333333333333 (pow l 3.0) (* l 2.0)) (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -2e+24) || !(t_0 <= 2e-6)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (J * (fma(0.3333333333333333, pow(l, 3.0), (l * 2.0)) * cos((K * 0.5))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -2e+24) || !(t_0 <= 2e-6))
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)) + U);
	else
		tmp = Float64(U + Float64(J * Float64(fma(0.3333333333333333, (l ^ 3.0), Float64(l * 2.0)) * cos(Float64(K * 0.5)))));
	end
	return 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, -2e+24], N[Not[LessEqual[t$95$0, 2e-6]], $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[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -2e24 or 1.99999999999999991e-6 < (-.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 -2e24 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 69.9%

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

   2. Taylor expanded in l around 0 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* 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -2 \cdot 10^{+24} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+24} \lor \neg \left(t_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -2e+24) (not (<= t_1 2e-6)))
     (+ (* t_0 (* t_1 J)) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -2e+24) || !(t_1 <= 2e-6)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-2d+24)) .or. (.not. (t_1 <= 2d-6))) then
        tmp = (t_0 * (t_1 * j)) + u
    else
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -2e+24) || !(t_1 <= 2e-6)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -2e+24) or not (t_1 <= 2e-6):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -2e+24) || !(t_1 <= 2e-6))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -2e+24) || ~((t_1 <= 2e-6)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+24], N[Not[LessEqual[t$95$1, 2e-6]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -2e24 or 1.99999999999999991e-6 < (-.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 -2e24 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 69.9%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -2 \cdot 10^{+24} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot {\ell}^{3}\\ t_1 := U + \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot t_0\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(t_0 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (pow l 3.0)))
        (t_1 (+ U (* (cos (* K 0.5)) (* J t_0))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -5e+125)
     t_1
     (if (<= l -57.0)
       t_2
       (if (<= l 16500000000.0)
         (+ U (* (cos (/ K 2.0)) (* J (+ t_0 (* l 2.0)))))
         (if (<= l 6.3e+100) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * pow(l, 3.0);
	double t_1 = U + (cos((K * 0.5)) * (J * t_0));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -5e+125) {
		tmp = t_1;
	} else if (l <= -57.0) {
		tmp = t_2;
	} else if (l <= 16500000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (t_0 + (l * 2.0))));
	} else if (l <= 6.3e+100) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (l ** 3.0d0)
    t_1 = u + (cos((k * 0.5d0)) * (j * t_0))
    t_2 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-5d+125)) then
        tmp = t_1
    else if (l <= (-57.0d0)) then
        tmp = t_2
    else if (l <= 16500000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (t_0 + (l * 2.0d0))))
    else if (l <= 6.3d+100) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * Math.pow(l, 3.0);
	double t_1 = U + (Math.cos((K * 0.5)) * (J * t_0));
	double t_2 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -5e+125) {
		tmp = t_1;
	} else if (l <= -57.0) {
		tmp = t_2;
	} else if (l <= 16500000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (t_0 + (l * 2.0))));
	} else if (l <= 6.3e+100) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * math.pow(l, 3.0)
	t_1 = U + (math.cos((K * 0.5)) * (J * t_0))
	t_2 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -5e+125:
		tmp = t_1
	elif l <= -57.0:
		tmp = t_2
	elif l <= 16500000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (t_0 + (l * 2.0))))
	elif l <= 6.3e+100:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * (l ^ 3.0))
	t_1 = Float64(U + Float64(cos(Float64(K * 0.5)) * Float64(J * t_0)))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -5e+125)
		tmp = t_1;
	elseif (l <= -57.0)
		tmp = t_2;
	elseif (l <= 16500000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(t_0 + Float64(l * 2.0)))));
	elseif (l <= 6.3e+100)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (l ^ 3.0);
	t_1 = U + (cos((K * 0.5)) * (J * t_0));
	t_2 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -5e+125)
		tmp = t_1;
	elseif (l <= -57.0)
		tmp = t_2;
	elseif (l <= 16500000000.0)
		tmp = U + (cos((K / 2.0)) * (J * (t_0 + (l * 2.0))));
	elseif (l <= 6.3e+100)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+125], t$95$1, If[LessEqual[l, -57.0], t$95$2, If[LessEqual[l, 16500000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.3e+100], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.99999999999999962e125 or 6.3000000000000004e100 < 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 J around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -4.99999999999999962e125 < l < -57 or 1.65e10 < l < 6.3000000000000004e100

   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 79.6%

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

   if -57 < l < 1.65e10

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 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. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+125}:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+100}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + \left(\ell \cdot t_0\right) \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* t_0 (* J (* 0.3333333333333333 (pow l 3.0))))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -1.96e+105)
     t_1
     (if (<= l -57.0)
       t_2
       (if (<= l 16500000000.0)
         (+ U (* (* l t_0) (* J 2.0)))
         (if (<= l 6.5e+100) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (t_0 * (J * (0.3333333333333333 * pow(l, 3.0))));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -1.96e+105) {
		tmp = t_1;
	} else if (l <= -57.0) {
		tmp = t_2;
	} else if (l <= 16500000000.0) {
		tmp = U + ((l * t_0) * (J * 2.0));
	} else if (l <= 6.5e+100) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + (t_0 * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    t_2 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-1.96d+105)) then
        tmp = t_1
    else if (l <= (-57.0d0)) then
        tmp = t_2
    else if (l <= 16500000000.0d0) then
        tmp = u + ((l * t_0) * (j * 2.0d0))
    else if (l <= 6.5d+100) then
        tmp = t_2
    else
        tmp = t_1
    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 = U + (t_0 * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	double t_2 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -1.96e+105) {
		tmp = t_1;
	} else if (l <= -57.0) {
		tmp = t_2;
	} else if (l <= 16500000000.0) {
		tmp = U + ((l * t_0) * (J * 2.0));
	} else if (l <= 6.5e+100) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (t_0 * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	t_2 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -1.96e+105:
		tmp = t_1
	elif l <= -57.0:
		tmp = t_2
	elif l <= 16500000000.0:
		tmp = U + ((l * t_0) * (J * 2.0))
	elif l <= 6.5e+100:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(t_0 * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -1.96e+105)
		tmp = t_1;
	elseif (l <= -57.0)
		tmp = t_2;
	elseif (l <= 16500000000.0)
		tmp = Float64(U + Float64(Float64(l * t_0) * Float64(J * 2.0)));
	elseif (l <= 6.5e+100)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + (t_0 * (J * (0.3333333333333333 * (l ^ 3.0))));
	t_2 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -1.96e+105)
		tmp = t_1;
	elseif (l <= -57.0)
		tmp = t_2;
	elseif (l <= 16500000000.0)
		tmp = U + ((l * t_0) * (J * 2.0));
	elseif (l <= 6.5e+100)
		tmp = t_2;
	else
		tmp = t_1;
	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[(U + N[(t$95$0 * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.96e+105], t$95$1, If[LessEqual[l, -57.0], t$95$2, If[LessEqual[l, 16500000000.0], N[(U + N[(N[(l * t$95$0), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+100], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.95999999999999994e105 or 6.50000000000000001e100 < 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 J around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -1.95999999999999994e105 < l < -57 or 1.65e10 < l < 6.50000000000000001e100

   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 79.6%

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

   if -57 < l < 1.65e10

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 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 J around 0 98.3%

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

      1. associate-*r* 98.3%

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

      2. *-commutative 98.3%

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

      3. *-commutative 98.3%

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

      4. fma-def 98.3%

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

      5. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in l around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*r* 98.2%

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

      3. associate-*l* 98.2%

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

   8. Simplified 98.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+105}:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+100}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 5: 77.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.55)
   (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
   (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.55], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.4%

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

   2. Taylor expanded in l around 0 88.1%

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

   3. Taylor expanded in l around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*l* 75.1%

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

   5. Simplified 75.1%

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

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

   1. Initial program 90.8%

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

   2. Taylor expanded in K around 0 90.8%

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

   3. Taylor expanded in l around 0 81.7%

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

4. Final simplification 79.2%

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

Alternative 6: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.6 \cdot 10^{+143}:\\ \;\;\;\;U + \left(\ell \cdot t_0\right) \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;J \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= J -1.6e+143)
     (+ U (* (* l t_0) (* J 2.0)))
     (if (<= J 1.45e+73)
       (+ U (* (- (exp l) (exp (- l))) J))
       (+ U (* l (* J (* 2.0 t_0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (J <= -1.6e+143) {
		tmp = U + ((l * t_0) * (J * 2.0));
	} else if (J <= 1.45e+73) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = U + (l * (J * (2.0 * 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 = cos((k * 0.5d0))
    if (j <= (-1.6d+143)) then
        tmp = u + ((l * t_0) * (j * 2.0d0))
    else if (j <= 1.45d+73) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else
        tmp = u + (l * (j * (2.0d0 * 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.cos((K * 0.5));
	double tmp;
	if (J <= -1.6e+143) {
		tmp = U + ((l * t_0) * (J * 2.0));
	} else if (J <= 1.45e+73) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else {
		tmp = U + (l * (J * (2.0 * t_0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if J <= -1.6e+143:
		tmp = U + ((l * t_0) * (J * 2.0))
	elif J <= 1.45e+73:
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	else:
		tmp = U + (l * (J * (2.0 * t_0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (J <= -1.6e+143)
		tmp = Float64(U + Float64(Float64(l * t_0) * Float64(J * 2.0)));
	elseif (J <= 1.45e+73)
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (J <= -1.6e+143)
		tmp = U + ((l * t_0) * (J * 2.0));
	elseif (J <= 1.45e+73)
		tmp = U + ((exp(l) - exp(-l)) * J);
	else
		tmp = U + (l * (J * (2.0 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[J, -1.6e+143], N[(U + N[(N[(l * t$95$0), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.45e+73], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.60000000000000008e143

   1. Initial program 64.3%

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

   2. Taylor expanded in l around 0 97.8%

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

   3. Taylor expanded in J around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. *-commutative 97.8%

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

      4. fma-def 97.8%

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

      5. *-commutative 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in l around 0 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*r* 95.7%

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

      3. associate-*l* 95.7%

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

   8. Simplified 95.7%

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

   if -1.60000000000000008e143 < J < 1.4500000000000001e73

   1. Initial program 97.5%

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

   2. Taylor expanded in K around 0 86.8%

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

   if 1.4500000000000001e73 < J

   1. Initial program 75.0%

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

   2. Taylor expanded in l around 0 93.6%

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

   3. Taylor expanded in l around 0 82.4%

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

      1. associate-*r* 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-*l* 82.4%

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

   5. Simplified 82.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.6 \cdot 10^{+143}:\\ \;\;\;\;U + \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;J \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 64.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in l around 0 85.1%

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

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. associate-*r* 62.9%

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

   2. *-commutative 62.9%

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

   3. associate-*l* 62.8%

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

5. Simplified 62.8%

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

6. Final simplification 62.8%

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

Alternative 8: 64.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (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 + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in l around 0 62.9%

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

   1. associate-*r* 62.9%

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

   2. *-commutative 62.9%

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

   3. associate-*l* 62.9%

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

4. Simplified 62.9%

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

5. Final simplification 62.9%

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

Alternative 9: 64.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + ((l * cos((K * 0.5))) * (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 * cos((k * 0.5d0))) * (j * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in l around 0 85.1%

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

3. Taylor expanded in J around 0 85.1%

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

   1. associate-*r* 85.1%

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

   2. *-commutative 85.1%

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

   3. *-commutative 85.1%

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

   4. fma-def 85.1%

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

   5. *-commutative 85.1%

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

5. Simplified 85.1%

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

6. Taylor expanded in l around 0 62.9%

   \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)} + U \]
7. 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. associate-*r* 62.9%

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

   3. associate-*l* 62.9%

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

8. Simplified 62.9%

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

9. Final simplification 62.9%

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

Alternative 10: 60.2% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -56 \lor \neg \left(\ell \leq 260\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -56.0) (not (<= l 260.0)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -56.0) || !(l <= 260.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-56.0d0)) .or. (.not. (l <= 260.0d0))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -56.0) || !(l <= 260.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -56.0) or not (l <= 260.0):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -56.0) || !(l <= 260.0))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -56.0) || ~((l <= 260.0)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -56.0], N[Not[LessEqual[l, 260.0]], $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[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -56 or 260 < 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 34.1%

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

      1. associate-*r* 34.1%

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

      2. *-commutative 34.1%

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

      3. associate-*l* 34.1%

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

   4. Simplified 34.1%

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

   5. Taylor expanded in K around 0 13.1%

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

      1. +-commutative 13.1%

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

      2. associate-*r* 13.1%

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

      3. distribute-rgt-out 36.0%

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

      4. *-commutative 36.0%

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

      5. unpow2 36.0%

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

   7. Simplified 36.0%

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

   if -56 < l < 260

   1. Initial program 69.9%

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

   2. Taylor expanded in K around 0 69.4%

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

   3. Taylor expanded in l around 0 80.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -56 \lor \neg \left(\ell \leq 260\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 11: 41.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(U \cdot -134217728\right)\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.75 \cdot 10^{+60}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -56 \lor \neg \left(\ell \leq 1.35 \cdot 10^{+74}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (* U -134217728.0))))
   (if (<= l -3.2e+133)
     t_0
     (if (<= l -2.75e+60)
       (* U U)
       (if (or (<= l -56.0) (not (<= l 1.35e+74))) t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (U * -134217728.0);
	double tmp;
	if (l <= -3.2e+133) {
		tmp = t_0;
	} else if (l <= -2.75e+60) {
		tmp = U * U;
	} else if ((l <= -56.0) || !(l <= 1.35e+74)) {
		tmp = t_0;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u * (u * (-134217728.0d0))
    if (l <= (-3.2d+133)) then
        tmp = t_0
    else if (l <= (-2.75d+60)) then
        tmp = u * u
    else if ((l <= (-56.0d0)) .or. (.not. (l <= 1.35d+74))) then
        tmp = t_0
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U * (U * -134217728.0);
	double tmp;
	if (l <= -3.2e+133) {
		tmp = t_0;
	} else if (l <= -2.75e+60) {
		tmp = U * U;
	} else if ((l <= -56.0) || !(l <= 1.35e+74)) {
		tmp = t_0;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (U * -134217728.0)
	tmp = 0
	if l <= -3.2e+133:
		tmp = t_0
	elif l <= -2.75e+60:
		tmp = U * U
	elif (l <= -56.0) or not (l <= 1.35e+74):
		tmp = t_0
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(U * -134217728.0))
	tmp = 0.0
	if (l <= -3.2e+133)
		tmp = t_0;
	elseif (l <= -2.75e+60)
		tmp = Float64(U * U);
	elseif ((l <= -56.0) || !(l <= 1.35e+74))
		tmp = t_0;
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (U * -134217728.0);
	tmp = 0.0;
	if (l <= -3.2e+133)
		tmp = t_0;
	elseif (l <= -2.75e+60)
		tmp = U * U;
	elseif ((l <= -56.0) || ~((l <= 1.35e+74)))
		tmp = t_0;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(U * -134217728.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.2e+133], t$95$0, If[LessEqual[l, -2.75e+60], N[(U * U), $MachinePrecision], If[Or[LessEqual[l, -56.0], N[Not[LessEqual[l, 1.35e+74]], $MachinePrecision]], t$95$0, U]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.19999999999999997e133 or -2.75e60 < l < -56 or 1.3499999999999999e74 < 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 2.8%

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

   5. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\left(U \cdot -134217728\right) \cdot U} \]

   if -3.19999999999999997e133 < l < -2.75e60

   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 0.7%

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

   5. Applied egg-rr 37.3%

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

   if -56 < l < 1.3499999999999999e74

   1. Initial program 73.5%

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

   3. Applied egg-rr 43.9%

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

   4. Taylor expanded in J around 0 61.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+133}:\\ \;\;\;\;U \cdot \left(U \cdot -134217728\right)\\ \mathbf{elif}\;\ell \leq -2.75 \cdot 10^{+60}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -56 \lor \neg \left(\ell \leq 1.35 \cdot 10^{+74}\right):\\ \;\;\;\;U \cdot \left(U \cdot -134217728\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 12: 41.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+15}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.55e+15) (* U U) (if (<= l 6.2e-7) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e+15) {
		tmp = U * U;
	} else if (l <= 6.2e-7) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.55d+15)) then
        tmp = u * u
    else if (l <= 6.2d-7) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e+15) {
		tmp = U * U;
	} else if (l <= 6.2e-7) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.55e+15:
		tmp = U * U
	elif l <= 6.2e-7:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.55e+15)
		tmp = Float64(U * U);
	elseif (l <= 6.2e-7)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.55e+15)
		tmp = U * U;
	elseif (l <= 6.2e-7)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.55e+15], N[(U * U), $MachinePrecision], If[LessEqual[l, 6.2e-7], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.55e15 or 6.1999999999999999e-7 < l

   1. Initial program 99.8%

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

   3. Applied egg-rr 3.5%

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

   5. Applied egg-rr 12.6%

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

   if -1.55e15 < l < 6.1999999999999999e-7

   1. Initial program 70.7%

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

   3. Applied egg-rr 47.4%

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

   4. Taylor expanded in J around 0 68.1%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+15}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 13: 54.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in K around 0 72.6%

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

3. Taylor expanded in l around 0 48.3%

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

4. Final simplification 48.3%

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

Alternative 14: 36.5% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Applied egg-rr 23.1%

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

4. Taylor expanded in J around 0 31.6%

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

5. Final simplification 31.6%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023275 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))