Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.5%

Time: 9.9s

Alternatives: 16

Speedup: 1.4×

• 50.1% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       l
       (* (* J (cos (* K 0.5))) (fma 0.3333333333333333 (pow l 2.0) 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 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * ((J * cos((K * 0.5))) * fma(0.3333333333333333, pow(l, 2.0), 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 <= Float64(-Inf)) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(l * Float64(Float64(J * cos(Float64(K * 0.5))) * fma(0.3333333333333333, (l ^ 2.0), 2.0))));
	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, (-Infinity)], 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[(U + N[(l * N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 71.2%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-out 99.9%

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

      9. fma-define 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.0)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow 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 <= -((double) INFINITY)) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(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 <= -math.inf) or not (t_1 <= 0.0):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(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 <= Float64(-Inf)) || !(t_1 <= 0.0))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (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 <= -Inf) || ~((t_1 <= 0.0)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 71.2%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 90.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* 0.3333333333333333 (* J (pow l 3.0)))))))
   (if (<= l -1.55e+85)
     t_0
     (if (<= l -2.45e-5)
       (* (- (exp l) (exp (- l))) J)
       (if (<= l 5.6e-15) (+ U (* l (* (cos (* K 0.5)) (* J 2.0)))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (0.3333333333333333 * (J * pow(l, 3.0))));
	double tmp;
	if (l <= -1.55e+85) {
		tmp = t_0;
	} else if (l <= -2.45e-5) {
		tmp = (exp(l) - exp(-l)) * J;
	} else if (l <= 5.6e-15) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (0.3333333333333333d0 * (j * (l ** 3.0d0))))
    if (l <= (-1.55d+85)) then
        tmp = t_0
    else if (l <= (-2.45d-5)) then
        tmp = (exp(l) - exp(-l)) * j
    else if (l <= 5.6d-15) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (0.3333333333333333 * (J * Math.pow(l, 3.0))));
	double tmp;
	if (l <= -1.55e+85) {
		tmp = t_0;
	} else if (l <= -2.45e-5) {
		tmp = (Math.exp(l) - Math.exp(-l)) * J;
	} else if (l <= 5.6e-15) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (0.3333333333333333 * (J * math.pow(l, 3.0))))
	tmp = 0
	if l <= -1.55e+85:
		tmp = t_0
	elif l <= -2.45e-5:
		tmp = (math.exp(l) - math.exp(-l)) * J
	elif l <= 5.6e-15:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))))
	tmp = 0.0
	if (l <= -1.55e+85)
		tmp = t_0;
	elseif (l <= -2.45e-5)
		tmp = Float64(Float64(exp(l) - exp(Float64(-l))) * J);
	elseif (l <= 5.6e-15)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (0.3333333333333333 * (J * (l ^ 3.0))));
	tmp = 0.0;
	if (l <= -1.55e+85)
		tmp = t_0;
	elseif (l <= -2.45e-5)
		tmp = (exp(l) - exp(-l)) * J;
	elseif (l <= 5.6e-15)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.55e+85], t$95$0, If[LessEqual[l, -2.45e-5], N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], If[LessEqual[l, 5.6e-15], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.55000000000000006e85 or 5.60000000000000028e-15 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 91.8%

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

   4. Taylor expanded in l around inf 91.8%

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

   if -1.55000000000000006e85 < l < -2.45e-5

   1. Initial program 98.8%

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

   3. Taylor expanded in K around 0 75.2%

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

   4. Taylor expanded in J around inf 75.2%

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

   if -2.45e-5 < l < 5.60000000000000028e-15

   1. Initial program 71.1%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in l around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.8%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. *-commutative 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+85}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ U (* l (* (cos (* K 0.5)) (* J 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 tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 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) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 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 tmp;
	if (Math.cos((K / 2.0)) <= -0.005) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 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):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 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)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = U + (l * (cos((K * 0.5)) * (J * 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_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 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 #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 85.4%

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

   3. Taylor expanded in l around 0 88.6%

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

      1. distribute-lft-in 88.5%

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

      2. *-commutative 88.5%

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

      3. associate-*l* 88.5%

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

      4. unpow2 88.5%

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

      5. pow3 88.5%

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

   5. Applied egg-rr 88.5%

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

   6. Taylor expanded in l around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*r* 65.4%

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

      3. associate-*l* 65.4%

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

      4. *-commutative 65.4%

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

      5. associate-*r* 65.4%

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

      6. *-commutative 65.4%

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

   8. Simplified 65.4%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.1%

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

   3. Taylor expanded in l around 0 93.9%

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

      1. distribute-lft-in 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*l* 93.9%

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

      4. unpow2 93.9%

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

      5. pow3 93.9%

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

   5. Applied egg-rr 93.9%

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

   6. Taylor expanded in K around 0 91.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5} \lor \neg \left(\ell \leq 8.5\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.45e-5) (not (<= l 8.5)))
   (* (- (exp l) (exp (- l))) J)
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.45e-5) || !(l <= 8.5)) {
		tmp = (exp(l) - exp(-l)) * J;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.45d-5)) .or. (.not. (l <= 8.5d0))) then
        tmp = (exp(l) - exp(-l)) * j
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.45e-5) || !(l <= 8.5)) {
		tmp = (Math.exp(l) - Math.exp(-l)) * J;
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.45e-5) or not (l <= 8.5):
		tmp = (math.exp(l) - math.exp(-l)) * J
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.45e-5) || !(l <= 8.5))
		tmp = Float64(Float64(exp(l) - exp(Float64(-l))) * J);
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.45e-5) || ~((l <= 8.5)))
		tmp = (exp(l) - exp(-l)) * J;
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.45e-5], N[Not[LessEqual[l, 8.5]], $MachinePrecision]], N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.45e-5 or 8.5 < l

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0 76.4%

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

   4. Taylor expanded in J around inf 76.4%

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

   if -2.45e-5 < l < 8.5

   1. Initial program 72.0%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in l around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. *-commutative 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5} \lor \neg \left(\ell \leq 8.5\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * pow(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 * (l * (2.0d0 + (0.3333333333333333d0 * (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 * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in l around 0 92.5%

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

4. Final simplification 92.5%

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

Alternative 7: 77.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+74}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.45e-5)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 2.6e+74)
     (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))
     (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 2.6e+74) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	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 <= (-2.45d-5)) then
        tmp = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    else if (l <= 2.6d+74) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 2.6e+74) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.45e-5:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	elif l <= 2.6e+74:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.45e-5)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 2.6e+74)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.45e-5)
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	elseif (l <= 2.6e+74)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.45e-5], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+74], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 84.4%

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

      1. distribute-lft-in 84.4%

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

      2. *-commutative 84.4%

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

      3. associate-*l* 84.4%

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

      4. unpow2 84.4%

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

      5. pow3 84.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in K around 0 64.8%

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

   7. Taylor expanded in U around 0 64.6%

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

   if -2.45e-5 < l < 2.6000000000000001e74

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0 93.7%

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

      1. distribute-lft-in 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-*l* 93.7%

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

      4. unpow2 93.7%

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

      5. pow3 93.7%

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

   5. Applied egg-rr 93.7%

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

   6. Taylor expanded in l around 0 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*r* 92.9%

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

      3. associate-*l* 92.9%

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

      4. *-commutative 92.9%

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

      5. associate-*r* 92.9%

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

      6. *-commutative 92.9%

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

   8. Simplified 92.9%

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

   if 2.6000000000000001e74 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

   5. Taylor expanded in K around 0 84.3%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+74}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+73}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.45e-5)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 2.6e+73)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 2.6e+73) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	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 <= (-2.45d-5)) then
        tmp = j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))
    else if (l <= 2.6d+73) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 2.6e+73) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.45e-5:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	elif l <= 2.6e+73:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.45e-5)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 2.6e+73)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.45e-5)
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	elseif (l <= 2.6e+73)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.45e-5], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+73], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 84.4%

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

      1. distribute-lft-in 84.4%

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

      2. *-commutative 84.4%

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

      3. associate-*l* 84.4%

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

      4. unpow2 84.4%

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

      5. pow3 84.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in K around 0 64.8%

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

   7. Taylor expanded in U around 0 64.6%

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

   if -2.45e-5 < l < 2.6000000000000001e73

   1. Initial program 74.0%

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

   3. Taylor expanded in l around 0 92.8%

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

   if 2.6000000000000001e73 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

   5. Taylor expanded in K around 0 84.3%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+73}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+19} \lor \neg \left(\ell \leq 1.35 \cdot 10^{-28}\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.02e+19) (not (<= l 1.35e-28)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (fma (* J 2.0) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.02e+19) || !(l <= 1.35e-28)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = fma((J * 2.0), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.02e+19) || !(l <= 1.35e-28))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = fma(Float64(J * 2.0), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.02e+19], N[Not[LessEqual[l, 1.35e-28]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.02e19 or 1.3499999999999999e-28 < l

   1. Initial program 99.3%

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

   3. Taylor expanded in l around 0 87.5%

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

   4. Taylor expanded in l around inf 86.8%

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

   5. Taylor expanded in K around 0 69.5%

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

      1. associate-*r* 69.5%

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

      2. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   if -1.02e19 < l < 1.3499999999999999e-28

   1. Initial program 72.4%

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

   3. Taylor expanded in l around 0 97.6%

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

      1. distribute-lft-in 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*l* 97.6%

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

      4. unpow2 97.6%

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

      5. pow3 97.6%

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

   5. Applied egg-rr 97.6%

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

   6. Taylor expanded in K around 0 85.6%

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

   7. Taylor expanded in l around 0 85.3%

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

      1. +-commutative 85.3%

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

      2. associate-*r* 85.3%

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

      3. fma-define 85.3%

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

      4. *-commutative 85.3%

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

   9. Simplified 85.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+19} \lor \neg \left(\ell \leq 1.35 \cdot 10^{-28}\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.45e-5)
   (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
   (if (<= l 1.35e-28)
     (fma (* J 2.0) l U)
     (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0));
	} else if (l <= 1.35e-28) {
		tmp = fma((J * 2.0), l, U);
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.45e-5)
		tmp = Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)));
	elseif (l <= 1.35e-28)
		tmp = fma(Float64(J * 2.0), l, U);
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.45e-5], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e-28], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in l around 0 84.4%

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

      1. distribute-lft-in 84.4%

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

      2. *-commutative 84.4%

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

      3. associate-*l* 84.4%

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

      4. unpow2 84.4%

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

      5. pow3 84.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in K around 0 64.8%

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

   7. Taylor expanded in U around 0 64.6%

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

   if -2.45e-5 < l < 1.3499999999999999e-28

   1. Initial program 71.6%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in K around 0 87.4%

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

   7. Taylor expanded in l around 0 87.4%

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

      1. +-commutative 87.4%

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

      2. associate-*r* 87.4%

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

      3. fma-define 87.5%

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

      4. *-commutative 87.5%

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

   9. Simplified 87.5%

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

   if 1.3499999999999999e-28 < l

   1. Initial program 98.6%

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

   3. Taylor expanded in l around 0 87.1%

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

   4. Taylor expanded in l around inf 85.7%

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

   5. Taylor expanded in K around 0 71.7%

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

      1. associate-*r* 71.7%

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

      2. *-commutative 71.7%

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

   7. Simplified 71.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in l around 0 92.5%

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

   1. distribute-lft-in 92.5%

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

   2. *-commutative 92.5%

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

   3. associate-*l* 92.5%

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

   4. unpow2 92.5%

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

   5. pow3 92.5%

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

5. Applied egg-rr 92.5%

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

6. Taylor expanded in K around 0 77.5%

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

7. Taylor expanded in l around 0 57.3%

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

   1. +-commutative 57.3%

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

   2. associate-*r* 57.3%

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

   3. fma-define 57.3%

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

   4. *-commutative 57.3%

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

9. Simplified 57.3%

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

Alternative 12: 41.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 270000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.45e-5) (* U U) (if (<= l 270000000000.0) U (* U (+ U -1.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = U * U;
	} else if (l <= 270000000000.0) {
		tmp = U;
	} else {
		tmp = U * (U + -1.0);
	}
	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 <= (-2.45d-5)) then
        tmp = u * u
    else if (l <= 270000000000.0d0) then
        tmp = u
    else
        tmp = u * (u + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45e-5) {
		tmp = U * U;
	} else if (l <= 270000000000.0) {
		tmp = U;
	} else {
		tmp = U * (U + -1.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.45e-5:
		tmp = U * U
	elif l <= 270000000000.0:
		tmp = U
	else:
		tmp = U * (U + -1.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.45e-5)
		tmp = Float64(U * U);
	elseif (l <= 270000000000.0)
		tmp = U;
	else
		tmp = Float64(U * Float64(U + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.45e-5)
		tmp = U * U;
	elseif (l <= 270000000000.0)
		tmp = U;
	else
		tmp = U * (U + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.45e-5], N[(U * U), $MachinePrecision], If[LessEqual[l, 270000000000.0], U, N[(U * N[(U + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-5

   1. Initial program 99.7%

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

   4. Applied egg-rr 11.8%

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

   if -2.45e-5 < l < 2.7e11

   1. Initial program 72.2%

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

   3. Taylor expanded in J around 0 70.5%

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

   if 2.7e11 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 87.4%

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

      1. distribute-lft-in 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l* 87.4%

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

      4. unpow2 87.4%

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

      5. pow3 87.4%

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

   5. Applied egg-rr 87.4%

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

   7. Applied egg-rr 13.4%

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

      1. fma-undefine 13.4%

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

      2. neg-mul-1 13.4%

         \[\leadsto U \cdot U + \color{blue}{-1 \cdot U} \]

      3. distribute-rgt-out 13.4%

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

   9. Simplified 13.4%

      \[\leadsto \color{blue}{U \cdot \left(U + -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 41.0% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5} \lor \neg \left(\ell \leq 4.3 \cdot 10^{+68}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.45e-5) (not (<= l 4.3e+68))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.45e-5) || !(l <= 4.3e+68)) {
		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 <= (-2.45d-5)) .or. (.not. (l <= 4.3d+68))) 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 <= -2.45e-5) || !(l <= 4.3e+68)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.45e-5) or not (l <= 4.3e+68):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.45e-5) || !(l <= 4.3e+68))
		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 <= -2.45e-5) || ~((l <= 4.3e+68)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.45e-5], N[Not[LessEqual[l, 4.3e+68]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2.45e-5 or 4.3000000000000001e68 < l

   1. Initial program 99.8%

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

   4. Applied egg-rr 13.3%

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

   if -2.45e-5 < l < 4.3000000000000001e68

   1. Initial program 73.8%

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

   3. Taylor expanded in J around 0 66.5%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-5} \lor \neg \left(\ell \leq 4.3 \cdot 10^{+68}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in K around 0 73.9%

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

4. Taylor expanded in l around 0 57.3%

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

   1. associate-*r* 57.3%

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

6. Simplified 57.3%

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

7. Final simplification 57.3%

   \[\leadsto U + \ell \cdot \left(J \cdot 2\right) \]
8. Add Preprocessing

Alternative 15: 36.1% 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 85.9%

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

3. Taylor expanded in J around 0 36.6%

   \[\leadsto \color{blue}{U} \]
4. Add Preprocessing

Alternative 16: 2.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

4. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

6. Simplified 2.8%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))