Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.5%

Time: 10.6s

Alternatives: 16

Speedup: 1.4×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+
      U
      (*
       J
       (* (cos (* K 0.5)) (fma 0.3333333333333333 (pow l 3.0) (* 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 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (J * (cos((K * 0.5)) * fma(0.3333333333333333, pow(l, 3.0), (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 <= Float64(-Inf)) || !(t_0 <= 0.0))
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)) + U);
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * fma(0.3333333333333333, (l ^ 3.0), Float64(l * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.8%

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

      1. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*r* 99.9%

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

      10. distribute-rgt-out 99.9%

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

      11. +-commutative 99.9%

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

      12. fma-def 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.8%

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

4. Final simplification 99.9%

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

Alternative 3: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+102} \lor \neg \left(\ell \leq -960000000000\right) \land \left(\ell \leq 6.2 \cdot 10^{-22} \lor \neg \left(\ell \leq 1.18 \cdot 10^{+91}\right)\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.8e+102)
         (and (not (<= l -960000000000.0))
              (or (<= l 6.2e-22) (not (<= l 1.18e+91)))))
   (+
    U
    (* (cos (/ K 2.0)) (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))
   (+ U (* (- (exp l) (exp (- l))) J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.8e+102) || (!(l <= -960000000000.0) && ((l <= 6.2e-22) || !(l <= 1.18e+91)))) {
		tmp = U + (cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))));
	} else {
		tmp = U + ((exp(l) - exp(-l)) * J);
	}
	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 <= (-5.8d+102)) .or. (.not. (l <= (-960000000000.0d0))) .and. (l <= 6.2d-22) .or. (.not. (l <= 1.18d+91))) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
    else
        tmp = u + ((exp(l) - exp(-l)) * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.8e+102) || (!(l <= -960000000000.0) && ((l <= 6.2e-22) || !(l <= 1.18e+91)))) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))));
	} else {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.8e+102) or (not (l <= -960000000000.0) and ((l <= 6.2e-22) or not (l <= 1.18e+91))):
		tmp = U + (math.cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))))
	else:
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.8e+102) || (!(l <= -960000000000.0) && ((l <= 6.2e-22) || !(l <= 1.18e+91))))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0))))));
	else
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -5.8e+102) || (~((l <= -960000000000.0)) && ((l <= 6.2e-22) || ~((l <= 1.18e+91)))))
		tmp = U + (cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))));
	else
		tmp = U + ((exp(l) - exp(-l)) * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.8e+102], And[N[Not[LessEqual[l, -960000000000.0]], $MachinePrecision], Or[LessEqual[l, 6.2e-22], N[Not[LessEqual[l, 1.18e+91]], $MachinePrecision]]]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.8000000000000005e102 or -9.6e11 < l < 6.20000000000000025e-22 or 1.18000000000000008e91 < l

   1. Initial program 80.7%

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

   2. Taylor expanded in l around 0 99.0%

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

   if -5.8000000000000005e102 < l < -9.6e11 or 6.20000000000000025e-22 < l < 1.18000000000000008e91

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 79.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+102} \lor \neg \left(\ell \leq -960000000000\right) \land \left(\ell \leq 6.2 \cdot 10^{-22} \lor \neg \left(\ell \leq 1.18 \cdot 10^{+91}\right)\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]

Alternative 4: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -960000000000:\\ \;\;\;\;U + t_1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-22} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+100}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + t_1 \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0)))))))
        (t_1 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -5.2e+102)
     t_0
     (if (<= l -960000000000.0)
       (+ U t_1)
       (if (or (<= l 6.2e-22) (not (<= l 1.7e+100)))
         t_0
         (+ U (* t_1 (+ (* -0.125 (* K K)) 1.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))));
	double t_1 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -5.2e+102) {
		tmp = t_0;
	} else if (l <= -960000000000.0) {
		tmp = U + t_1;
	} else if ((l <= 6.2e-22) || !(l <= 1.7e+100)) {
		tmp = t_0;
	} else {
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
    t_1 = (exp(l) - exp(-l)) * j
    if (l <= (-5.2d+102)) then
        tmp = t_0
    else if (l <= (-960000000000.0d0)) then
        tmp = u + t_1
    else if ((l <= 6.2d-22) .or. (.not. (l <= 1.7d+100))) then
        tmp = t_0
    else
        tmp = u + (t_1 * (((-0.125d0) * (k * k)) + 1.0d0))
    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)) * (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))));
	double t_1 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -5.2e+102) {
		tmp = t_0;
	} else if (l <= -960000000000.0) {
		tmp = U + t_1;
	} else if ((l <= 6.2e-22) || !(l <= 1.7e+100)) {
		tmp = t_0;
	} else {
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))))
	t_1 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -5.2e+102:
		tmp = t_0
	elif l <= -960000000000.0:
		tmp = U + t_1
	elif (l <= 6.2e-22) or not (l <= 1.7e+100):
		tmp = t_0
	else:
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0))))))
	t_1 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -5.2e+102)
		tmp = t_0;
	elseif (l <= -960000000000.0)
		tmp = Float64(U + t_1);
	elseif ((l <= 6.2e-22) || !(l <= 1.7e+100))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(t_1 * Float64(Float64(-0.125 * Float64(K * K)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))));
	t_1 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -5.2e+102)
		tmp = t_0;
	elseif (l <= -960000000000.0)
		tmp = U + t_1;
	elseif ((l <= 6.2e-22) || ~((l <= 1.7e+100)))
		tmp = t_0;
	else
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0));
	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[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -5.2e+102], t$95$0, If[LessEqual[l, -960000000000.0], N[(U + t$95$1), $MachinePrecision], If[Or[LessEqual[l, 6.2e-22], N[Not[LessEqual[l, 1.7e+100]], $MachinePrecision]], t$95$0, N[(U + N[(t$95$1 * N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.20000000000000013e102 or -9.6e11 < l < 6.20000000000000025e-22 or 1.69999999999999997e100 < l

   1. Initial program 80.6%

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

   2. Taylor expanded in l around 0 99.5%

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

   if -5.20000000000000013e102 < l < -9.6e11

   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 80.0%

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

   if 6.20000000000000025e-22 < l < 1.69999999999999997e100

   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 12.0%

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + -0.125 \cdot \left({K}^{2} \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 12.0%

         \[\leadsto \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + \color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)}\right) + U \]

      2. distribute-rgt1-in 84.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow2 84.0%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 84.0%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -960000000000:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-22} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+100}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right)\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -105000000000 \lor \neg \left(\ell \leq 6.2 \cdot 10^{-22}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -105000000000.0) (not (<= l 6.2e-22)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (fma (* (cos (* K 0.5)) (* J 2.0)) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -105000000000.0) || !(l <= 6.2e-22)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = fma((cos((K * 0.5)) * (J * 2.0)), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -105000000000.0) || !(l <= 6.2e-22))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = fma(Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0)), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -105000000000.0], N[Not[LessEqual[l, 6.2e-22]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.05e11 or 6.20000000000000025e-22 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 76.9%

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

   if -1.05e11 < l < 6.20000000000000025e-22

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r* 99.0%

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

      3. associate-*r* 99.1%

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

      4. fma-def 99.1%

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

      5. div-inv 99.1%

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

      6. metadata-eval 99.1%

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

      7. *-commutative 99.1%

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

   4. Applied egg-rr 99.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -105000000000 \lor \neg \left(\ell \leq 6.2 \cdot 10^{-22}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \end{array} \]

Alternative 6: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -105000000000 \lor \neg \left(\ell \leq 6.2 \cdot 10^{-22}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -105000000000.0) (not (<= l 6.2e-22)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -105000000000.0) || !(l <= 6.2e-22)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-105000000000.0d0)) .or. (.not. (l <= 6.2d-22))) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -105000000000.0) || !(l <= 6.2e-22)) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -105000000000.0) or not (l <= 6.2e-22):
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -105000000000.0) || !(l <= 6.2e-22))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -105000000000.0) || ~((l <= 6.2e-22)))
		tmp = U + ((exp(l) - exp(-l)) * J);
	else
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -105000000000.0], N[Not[LessEqual[l, 6.2e-22]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.05e11 or 6.20000000000000025e-22 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 76.9%

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

   if -1.05e11 < l < 6.20000000000000025e-22

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 99.1%

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

      1. associate-*r* 99.2%

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

      2. associate-*r* 99.2%

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

      3. associate-*r* 99.2%

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

      4. associate-*r* 99.2%

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

      5. distribute-rgt-out 99.2%

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

      6. *-commutative 99.2%

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

      7. associate-*r* 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.2%

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

      10. distribute-rgt-out 99.2%

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

      11. +-commutative 99.2%

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

      12. fma-def 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -105000000000 \lor \neg \left(\ell \leq 6.2 \cdot 10^{-22}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 7: 78.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -320000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 6500:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+88}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -320000000000.0)
     t_0
     (if (<= l 6500.0)
       (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
       (if (<= l 2.15e+88) (pow U -8.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -320000000000.0) {
		tmp = t_0;
	} else if (l <= 6500.0) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 2.15e+88) {
		tmp = pow(U, -8.0);
	} 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 + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-320000000000.0d0)) then
        tmp = t_0
    else if (l <= 6500.0d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (l <= 2.15d+88) then
        tmp = u ** (-8.0d0)
    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 + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -320000000000.0) {
		tmp = t_0;
	} else if (l <= 6500.0) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 2.15e+88) {
		tmp = Math.pow(U, -8.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -320000000000.0:
		tmp = t_0
	elif l <= 6500.0:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 2.15e+88:
		tmp = math.pow(U, -8.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -320000000000.0)
		tmp = t_0;
	elseif (l <= 6500.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 2.15e+88)
		tmp = U ^ -8.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -320000000000.0)
		tmp = t_0;
	elseif (l <= 6500.0)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (l <= 2.15e+88)
		tmp = U ^ -8.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -320000000000.0], t$95$0, If[LessEqual[l, 6500.0], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.15e+88], N[Power[U, -8.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2e11 or 2.14999999999999987e88 < 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 86.2%

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

      1. associate-*r* 86.2%

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

      2. associate-*r* 86.2%

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

      3. associate-*r* 86.2%

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

      4. associate-*r* 86.2%

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

      5. distribute-rgt-out 86.2%

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

      6. *-commutative 86.2%

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

      7. associate-*r* 86.2%

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

      8. *-commutative 86.2%

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

      9. associate-*r* 86.2%

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

      10. distribute-rgt-out 86.2%

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

      11. +-commutative 86.2%

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

      12. fma-def 86.2%

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

   4. Simplified 86.2%

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

   5. Taylor expanded in K around 0 67.5%

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

   6. Taylor expanded in l around inf 67.5%

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

   if -3.2e11 < l < 6500

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 97.8%

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

      1. associate-*r* 97.9%

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

      2. associate-*r* 97.9%

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

      3. associate-*r* 97.9%

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

      4. associate-*r* 97.9%

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

      5. distribute-rgt-out 97.9%

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

      6. *-commutative 97.9%

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

      7. associate-*r* 97.9%

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

      8. *-commutative 97.9%

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

      9. associate-*r* 97.9%

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

      10. distribute-rgt-out 97.9%

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

      11. +-commutative 97.9%

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

      12. fma-def 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in l around 0 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*r* 97.8%

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

   7. Simplified 97.8%

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

   if 6500 < l < 2.14999999999999987e88

   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 35.6%

      \[\leadsto \color{blue}{{U}^{-8}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -320000000000:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 6500:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+88}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 8: 78.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 6500:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.38 \cdot 10^{+88}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.9e-6)
   (+ U (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0)))))
   (if (<= l 6500.0)
     (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
     (if (<= l 1.38e+88)
       (pow U -8.0)
       (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.9e-6) {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0))));
	} else if (l <= 6500.0) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 1.38e+88) {
		tmp = pow(U, -8.0);
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	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 <= (-1.9d-6)) then
        tmp = u + (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0))))
    else if (l <= 6500.0d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (l <= 1.38d+88) then
        tmp = u ** (-8.0d0)
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    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-6) {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0))));
	} else if (l <= 6500.0) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 1.38e+88) {
		tmp = Math.pow(U, -8.0);
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.9e-6:
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0))))
	elif l <= 6500.0:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 1.38e+88:
		tmp = math.pow(U, -8.0)
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.9e-6)
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))));
	elseif (l <= 6500.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 1.38e+88)
		tmp = U ^ -8.0;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.9e-6)
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0))));
	elseif (l <= 6500.0)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (l <= 1.38e+88)
		tmp = U ^ -8.0;
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.9e-6], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6500.0], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.38e+88], N[Power[U, -8.0], $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.9e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in l around 0 77.0%

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

      1. associate-*r* 77.0%

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

      2. associate-*r* 77.0%

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

      3. associate-*r* 77.0%

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

      4. associate-*r* 77.0%

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

      5. distribute-rgt-out 77.0%

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

      6. *-commutative 77.0%

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

      7. associate-*r* 77.0%

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

      8. *-commutative 77.0%

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

      9. associate-*r* 77.0%

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

      10. distribute-rgt-out 77.0%

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

      11. +-commutative 77.0%

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

      12. fma-def 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in K around 0 57.7%

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

   if -1.9e-6 < l < 6500

   1. Initial program 70.1%

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

   2. Taylor expanded in l around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. associate-*r* 98.5%

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

      3. associate-*r* 98.5%

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

      4. associate-*r* 98.5%

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

      5. distribute-rgt-out 98.5%

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

      6. *-commutative 98.5%

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

      7. associate-*r* 98.5%

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

      8. *-commutative 98.5%

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

      9. associate-*r* 98.5%

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

      10. distribute-rgt-out 98.5%

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

      11. +-commutative 98.5%

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

      12. fma-def 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in l around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

   7. Simplified 98.5%

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

   if 6500 < l < 1.37999999999999993e88

   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 35.6%

      \[\leadsto \color{blue}{{U}^{-8}} \]

   if 1.37999999999999993e88 < 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 97.7%

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

      1. associate-*r* 97.7%

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

      2. associate-*r* 97.7%

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

      3. associate-*r* 97.7%

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

      4. associate-*r* 97.7%

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

      5. distribute-rgt-out 97.7%

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

      6. *-commutative 97.7%

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

      7. associate-*r* 97.7%

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

      8. *-commutative 97.7%

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

      9. associate-*r* 97.7%

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

      10. distribute-rgt-out 97.7%

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

      11. +-commutative 97.7%

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

      12. fma-def 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in K around 0 81.0%

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

   6. Taylor expanded in l around inf 81.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 6500:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.38 \cdot 10^{+88}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 9: 72.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -105000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 6500:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -105000000000.0)
     t_0
     (if (<= l 6500.0)
       (fma l (* J 2.0) U)
       (if (<= l 1.4e+89) (pow U -8.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -105000000000.0) {
		tmp = t_0;
	} else if (l <= 6500.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 1.4e+89) {
		tmp = pow(U, -8.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -105000000000.0)
		tmp = t_0;
	elseif (l <= 6500.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 1.4e+89)
		tmp = U ^ -8.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -105000000000.0], t$95$0, If[LessEqual[l, 6500.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.4e+89], N[Power[U, -8.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.05e11 or 1.3999999999999999e89 < 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 86.2%

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

      1. associate-*r* 86.2%

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

      2. associate-*r* 86.2%

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

      3. associate-*r* 86.2%

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

      4. associate-*r* 86.2%

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

      5. distribute-rgt-out 86.2%

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

      6. *-commutative 86.2%

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

      7. associate-*r* 86.2%

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

      8. *-commutative 86.2%

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

      9. associate-*r* 86.2%

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

      10. distribute-rgt-out 86.2%

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

      11. +-commutative 86.2%

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

      12. fma-def 86.2%

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

   4. Simplified 86.2%

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

   5. Taylor expanded in K around 0 67.5%

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

   6. Taylor expanded in l around inf 67.5%

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

   if -1.05e11 < l < 6500

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 97.7%

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

   3. Taylor expanded in K around 0 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r* 86.4%

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

      3. fma-def 86.4%

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

      4. *-commutative 86.4%

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

   5. Simplified 86.4%

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

   if 6500 < l < 1.3999999999999999e89

   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 35.6%

      \[\leadsto \color{blue}{{U}^{-8}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -105000000000:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 6500:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 10: 57.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -760:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 11200:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+188}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -760.0)
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (if (<= l 11200.0)
     (fma l (* J 2.0) U)
     (if (<= l 5.2e+188) (pow U -8.0) (+ U (* l (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -760.0) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if (l <= 11200.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 5.2e+188) {
		tmp = pow(U, -8.0);
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -760.0)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	elseif (l <= 11200.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 5.2e+188)
		tmp = U ^ -8.0;
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -760.0], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 11200.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 5.2e+188], N[Power[U, -8.0], $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -760

   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 21.6%

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

   3. Taylor expanded in K around 0 18.1%

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

      1. +-commutative 18.1%

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

      2. associate-*r* 18.1%

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

      3. distribute-rgt-out 34.1%

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

      4. *-commutative 34.1%

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

      5. unpow2 34.1%

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

   5. Simplified 34.1%

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

   if -760 < l < 11200

   1. Initial program 70.3%

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

   2. Taylor expanded in l around 0 98.4%

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

   3. Taylor expanded in K around 0 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r* 87.0%

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

      3. fma-def 87.0%

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

      4. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   if 11200 < l < 5.19999999999999975e188

   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 42.7%

      \[\leadsto \color{blue}{{U}^{-8}} \]

   if 5.19999999999999975e188 < 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 76.4%

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

   3. Taylor expanded in K around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-*r* 67.8%

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

      3. *-commutative 67.8%

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

   5. Simplified 67.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -760:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 11200:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+188}:\\ \;\;\;\;{U}^{-8}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 11: 57.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -450:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 6500 \lor \neg \left(\ell \leq 2.55 \cdot 10^{+184}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -450.0)
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (if (or (<= l 6500.0) (not (<= l 2.55e+184)))
     (+ U (* l (* J 2.0)))
     (pow U -8.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -450.0) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if ((l <= 6500.0) || !(l <= 2.55e+184)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = pow(U, -8.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-450.0d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else if ((l <= 6500.0d0) .or. (.not. (l <= 2.55d+184))) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u ** (-8.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -450.0) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else if ((l <= 6500.0) || !(l <= 2.55e+184)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = Math.pow(U, -8.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -450.0:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	elif (l <= 6500.0) or not (l <= 2.55e+184):
		tmp = U + (l * (J * 2.0))
	else:
		tmp = math.pow(U, -8.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -450.0)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	elseif ((l <= 6500.0) || !(l <= 2.55e+184))
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = U ^ -8.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -450.0)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	elseif ((l <= 6500.0) || ~((l <= 2.55e+184)))
		tmp = U + (l * (J * 2.0));
	else
		tmp = U ^ -8.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -450.0], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 6500.0], N[Not[LessEqual[l, 2.55e+184]], $MachinePrecision]], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -8.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -450

   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 21.6%

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

   3. Taylor expanded in K around 0 18.1%

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

      1. +-commutative 18.1%

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

      2. associate-*r* 18.1%

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

      3. distribute-rgt-out 34.1%

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

      4. *-commutative 34.1%

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

      5. unpow2 34.1%

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

   5. Simplified 34.1%

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

   if -450 < l < 6500 or 2.5500000000000001e184 < l

   1. Initial program 74.7%

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

   2. Taylor expanded in l around 0 95.1%

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

   3. Taylor expanded in K around 0 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r* 84.2%

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

      3. *-commutative 84.2%

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

   5. Simplified 84.2%

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

   if 6500 < l < 2.5500000000000001e184

   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 42.7%

      \[\leadsto \color{blue}{{U}^{-8}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -450:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 6500 \lor \neg \left(\ell \leq 2.55 \cdot 10^{+184}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \]

Alternative 12: 58.6% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -300 \lor \neg \left(\ell \leq 540000\right) \land \ell \leq 5.5 \cdot 10^{+207}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -300.0) (and (not (<= l 540000.0)) (<= l 5.5e+207)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -300.0) || (!(l <= 540000.0) && (l <= 5.5e+207))) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-300.0d0)) .or. (.not. (l <= 540000.0d0)) .and. (l <= 5.5d+207)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (l * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -300.0) || (!(l <= 540000.0) && (l <= 5.5e+207))) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -300.0) or (not (l <= 540000.0) and (l <= 5.5e+207)):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -300.0) || (!(l <= 540000.0) && (l <= 5.5e+207)))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -300.0) || (~((l <= 540000.0)) && (l <= 5.5e+207)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -300.0], And[N[Not[LessEqual[l, 540000.0]], $MachinePrecision], LessEqual[l, 5.5e+207]]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -300 or 5.4e5 < l < 5.50000000000000036e207

   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 19.6%

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

   3. Taylor expanded in K around 0 16.3%

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

      1. +-commutative 16.3%

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

      2. associate-*r* 16.3%

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

      3. distribute-rgt-out 31.5%

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

      4. *-commutative 31.5%

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

      5. unpow2 31.5%

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

   5. Simplified 31.5%

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

   if -300 < l < 5.4e5 or 5.50000000000000036e207 < l

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 96.1%

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

   3. Taylor expanded in K around 0 86.1%

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

      1. *-commutative 86.1%

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

      2. associate-*r* 86.1%

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

      3. *-commutative 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -300 \lor \neg \left(\ell \leq 540000\right) \land \ell \leq 5.5 \cdot 10^{+207}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 13: 42.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 23:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.2e+18) (* U U) (if (<= l 23.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.2e+18) {
		tmp = U * U;
	} else if (l <= 23.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-8.2d+18)) then
        tmp = u * u
    else if (l <= 23.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.2e+18) {
		tmp = U * U;
	} else if (l <= 23.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -8.2e+18:
		tmp = U * U
	elif l <= 23.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.2e+18)
		tmp = Float64(U * U);
	elseif (l <= 23.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -8.2e+18)
		tmp = U * U;
	elseif (l <= 23.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.2e+18], N[(U * U), $MachinePrecision], If[LessEqual[l, 23.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -8.2e18 or 23 < 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 13.0%

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

   if -8.2e18 < l < 23

   1. Initial program 70.9%

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

   2. Taylor expanded in J around 0 66.8%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 23:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 14: 53.7% 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 84.0%

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

2. Taylor expanded in l around 0 66.6%

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

3. Taylor expanded in K around 0 57.9%

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

   1. *-commutative 57.9%

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

   2. associate-*r* 57.9%

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

   3. *-commutative 57.9%

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

5. Simplified 57.9%

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

6. Final simplification 57.9%

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

Alternative 15: 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 84.0%

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

3. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

5. Simplified 2.8%

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

6. Final simplification 2.8%

   \[\leadsto 1 \]

Alternative 16: 36.5% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

2. Taylor expanded in J around 0 37.7%

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

3. Final simplification 37.7%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023216 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))