Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 99.6%

Time: 10.9s

Alternatives: 12

Speedup: 1.5×

• 49.2% 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: 85.8% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 72.2%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 87.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -2.0) (not (<= t_0 0.0)))
     (+ (* t_0 J) U)
     (+ U (* (cos (/ K 2.0)) (* J (* 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 <= -2.0) || !(t_0 <= 0.0)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-2.0d0)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (t_0 * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (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 <= -2.0) || !(t_0 <= 0.0)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (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 <= -2.0) or not (t_0 <= 0.0):
		tmp = (t_0 * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (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 <= -2.0) || !(t_0 <= 0.0))
		tmp = Float64(Float64(t_0 * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * 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 <= -2.0) || ~((t_0 <= 0.0)))
		tmp = (t_0 * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * (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, -2.0], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * 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))) < -2 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 72.1%

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

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

   1. Initial program 72.2%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 86.7%

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

Alternative 3: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.00027:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.00019:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -7.6e+70)
     t_0
     (if (<= l -0.00027)
       t_1
       (if (<= l 0.00019)
         (fma (* l J) (* 2.0 (cos (* K 0.5))) U)
         (if (<= l 3.5e+82) 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 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -7.6e+70) {
		tmp = t_0;
	} else if (l <= -0.00027) {
		tmp = t_1;
	} else if (l <= 0.00019) {
		tmp = fma((l * J), (2.0 * cos((K * 0.5))), U);
	} else if (l <= 3.5e+82) {
		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(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -7.6e+70)
		tmp = t_0;
	elseif (l <= -0.00027)
		tmp = t_1;
	elseif (l <= 0.00019)
		tmp = fma(Float64(l * J), Float64(2.0 * cos(Float64(K * 0.5))), U);
	elseif (l <= 3.5e+82)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return 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[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -7.6e+70], t$95$0, If[LessEqual[l, -0.00027], t$95$1, If[LessEqual[l, 0.00019], N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 3.5e+82], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.5999999999999996e70 or 3.5e82 < 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 98.9%

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

   3. Taylor expanded in l around inf 98.9%

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

      1. associate-*r* 68.5%

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

   5. Simplified 98.9%

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

   if -7.5999999999999996e70 < l < -2.70000000000000003e-4 or 1.9000000000000001e-4 < l < 3.5e82

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 79.9%

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

   if -2.70000000000000003e-4 < l < 1.9000000000000001e-4

   1. Initial program 72.2%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+70}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00027:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.00019:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.055:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.055)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.055) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.055d0)) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.055) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.055:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.055)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.055)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.055], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0550000000000000003

   1. Initial program 86.5%

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

   2. Taylor expanded in l around 0 65.4%

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

   if -0.0550000000000000003 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.1%

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

   2. Taylor expanded in l around 0 90.8%

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

   3. Taylor expanded in K around 0 87.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.055:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 88.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (* (cos (/ K 2.0)) (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (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 + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
end

Wolfram

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

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in l around 0 91.8%

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

3. Final simplification 91.8%

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

Alternative 6: 87.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00265 \lor \neg \left(\ell \leq 0.00048\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.00265) (not (<= l 0.00048)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (fma (* l J) (* 2.0 (cos (* K 0.5))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00265) || !(l <= 0.00048)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = fma((l * J), (2.0 * cos((K * 0.5))), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.00265) || !(l <= 0.00048))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = fma(Float64(l * J), Float64(2.0 * cos(Float64(K * 0.5))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.00265], N[Not[LessEqual[l, 0.00048]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.00265000000000000001 or 4.80000000000000012e-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 72.1%

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

   if -0.00265000000000000001 < l < 4.80000000000000012e-4

   1. Initial program 72.2%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00265 \lor \neg \left(\ell \leq 0.00048\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \]

Alternative 7: 78.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+16} \lor \neg \left(\ell \leq 2.6 \cdot 10^{+66}\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.9e+16) (not (<= l 2.6e+66)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.9e+16) || !(l <= 2.6e+66)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	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 <= (-1.9d+16)) .or. (.not. (l <= 2.6d+66))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.9e+16) || !(l <= 2.6e+66)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.9e+16) or not (l <= 2.6e+66):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.9e+16) || !(l <= 2.6e+66))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.9e+16) || ~((l <= 2.6e+66)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.9e+16], N[Not[LessEqual[l, 2.6e+66]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.9e16 or 2.60000000000000012e66 < 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 90.7%

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

   3. Taylor expanded in K around 0 63.7%

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

   4. Taylor expanded in l around inf 63.7%

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

      1. associate-*r* 63.7%

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

   6. Simplified 63.7%

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

   if -1.9e16 < l < 2.60000000000000012e66

   1. Initial program 75.5%

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

   2. Taylor expanded in l around 0 91.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+16} \lor \neg \left(\ell \leq 2.6 \cdot 10^{+66}\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 8: 78.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+17} \lor \neg \left(\ell \leq 1.08 \cdot 10^{+67}\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.1e+17) (not (<= l 1.08e+67)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e+17) || !(l <= 1.08e+67)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (cos((K / 2.0)) * (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 <= (-2.1d+17)) .or. (.not. (l <= 1.08d+67))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    else
        tmp = u + (cos((k / 2.0d0)) * (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 <= -2.1e+17) || !(l <= 1.08e+67)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.1e+17) or not (l <= 1.08e+67):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.1e+17) || !(l <= 1.08e+67))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.1e+17) || ~((l <= 1.08e+67)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.1e+17], N[Not[LessEqual[l, 1.08e+67]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.1e17 or 1.08e67 < 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 90.7%

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

   3. Taylor expanded in K around 0 63.7%

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

   4. Taylor expanded in l around inf 63.7%

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

      1. associate-*r* 63.7%

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

   6. Simplified 63.7%

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

   if -2.1e17 < l < 1.08e67

   1. Initial program 75.5%

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

   2. Taylor expanded in l around 0 91.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+17} \lor \neg \left(\ell \leq 1.08 \cdot 10^{+67}\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 9: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -460000000000 \lor \neg \left(\ell \leq 0.00048\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -460000000000.0) (not (<= l 0.00048)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (fma l (* J 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -460000000000.0) || !(l <= 0.00048)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = fma(l, (J * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -460000000000.0) || !(l <= 0.00048))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = fma(l, Float64(J * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -460000000000.0], N[Not[LessEqual[l, 0.00048]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.6e11 or 4.80000000000000012e-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 l around 0 84.6%

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

   3. Taylor expanded in K around 0 59.4%

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

   4. Taylor expanded in l around inf 59.4%

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

      1. associate-*r* 59.4%

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

   6. Simplified 59.4%

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

   if -4.6e11 < l < 4.80000000000000012e-4

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 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. Taylor expanded in K around 0 86.2%

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

   4. Taylor expanded in l around 0 86.2%

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

      1. +-commutative 86.2%

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

      2. associate-*r* 86.2%

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

      3. *-commutative 86.2%

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

      4. fma-def 86.2%

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

      5. *-commutative 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -460000000000 \lor \neg \left(\ell \leq 0.00048\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \end{array} \]

Alternative 10: 42.5% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+30} \lor \neg \left(\ell \leq 240000000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.05e+30) (not (<= l 240000000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+30) || !(l <= 240000000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.05d+30)) .or. (.not. (l <= 240000000000.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+30) || !(l <= 240000000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.05e+30) or not (l <= 240000000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.05e+30) || !(l <= 240000000000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.05e+30) || ~((l <= 240000000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.05e+30], N[Not[LessEqual[l, 240000000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.05e30 or 2.4e11 < 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 14.6%

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

   if -1.05e30 < l < 2.4e11

   1. Initial program 73.8%

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

   2. Taylor expanded in J around 0 68.8%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+30} \lor \neg \left(\ell \leq 240000000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 11: 54.5% 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 85.5%

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

2. Taylor expanded in l around 0 91.8%

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

3. Taylor expanded in K around 0 73.7%

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

4. Taylor expanded in l around 0 58.7%

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

   1. *-commutative 58.7%

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

   2. associate-*r* 59.0%

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

   3. *-commutative 59.0%

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

6. Simplified 59.0%

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

7. Final simplification 59.0%

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

Alternative 12: 36.7% 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 85.5%

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

2. Taylor expanded in J around 0 39.1%

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

3. Final simplification 39.1%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023305 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))