Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.8%

Time: 12.3s

Alternatives: 14

Speedup: 1.4×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0.0005\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0005)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       J
       (*
        (cos (* K 0.5))
        (+ (* 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 <= 0.0005)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (cos((K * 0.5)) * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0005)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0005):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * ((0.3333333333333333 * math.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 <= 0.0005))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0005)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (J * (cos((K * 0.5)) * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 5.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-4

   1. Initial program 70.6%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. associate-*r* 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*r* 99.9%

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

      10. distribute-rgt-out 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*r* 99.9%

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

      14. *-commutative 99.9%

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

      15. distribute-rgt-in 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in J around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+201} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+206}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (or (<= t_0 -1e+201) (not (<= t_0 5e+206)))
     t_0
     (+ U (* l (* (cos (* K 0.5)) (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if ((t_0 <= -1e+201) || !(t_0 <= 5e+206)) {
		tmp = t_0;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    if ((t_0 <= (-1d+201)) .or. (.not. (t_0 <= 5d+206))) then
        tmp = t_0
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if ((t_0 <= -1e+201) || !(t_0 <= 5e+206)) {
		tmp = t_0;
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if (t_0 <= -1e+201) or not (t_0 <= 5e+206):
		tmp = t_0
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if ((t_0 <= -1e+201) || !(t_0 <= 5e+206))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if ((t_0 <= -1e+201) || ~((t_0 <= 5e+206)))
		tmp = t_0;
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+201], N[Not[LessEqual[t$95$0, 5e+206]], $MachinePrecision]], t$95$0, N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -1.00000000000000004e201 or 5.0000000000000002e206 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0 74.8%

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

   4. Taylor expanded in J around inf 74.8%

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

   if -1.00000000000000004e201 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 5.0000000000000002e206

   1. Initial program 70.7%

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

   3. Taylor expanded in l around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -1 \cdot 10^{+201} \lor \neg \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq 5 \cdot 10^{+206}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{5} \cdot \left(t\_0 \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -3.4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;U + \ell \cdot \left(t\_0 \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+52}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 5.0) (* t_0 (* J 0.016666666666666666))))))
   (if (<= l -3.4)
     t_1
     (if (<= l 3.3e-5)
       (+ U (* l (* t_0 (* J 2.0))))
       (if (<= l 7e+52) (+ (* (- (exp l) (exp (- l))) J) U) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -3.4) {
		tmp = t_1;
	} else if (l <= 3.3e-5) {
		tmp = U + (l * (t_0 * (J * 2.0)));
	} else if (l <= 7e+52) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((l ** 5.0d0) * (t_0 * (j * 0.016666666666666666d0)))
    if (l <= (-3.4d0)) then
        tmp = t_1
    else if (l <= 3.3d-5) then
        tmp = u + (l * (t_0 * (j * 2.0d0)))
    else if (l <= 7d+52) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -3.4) {
		tmp = t_1;
	} else if (l <= 3.3e-5) {
		tmp = U + (l * (t_0 * (J * 2.0)));
	} else if (l <= 7e+52) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)))
	tmp = 0
	if l <= -3.4:
		tmp = t_1
	elif l <= 3.3e-5:
		tmp = U + (l * (t_0 * (J * 2.0)))
	elif l <= 7e+52:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 5.0) * Float64(t_0 * Float64(J * 0.016666666666666666))))
	tmp = 0.0
	if (l <= -3.4)
		tmp = t_1;
	elseif (l <= 3.3e-5)
		tmp = Float64(U + Float64(l * Float64(t_0 * Float64(J * 2.0))));
	elseif (l <= 7e+52)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((l ^ 5.0) * (t_0 * (J * 0.016666666666666666)));
	tmp = 0.0;
	if (l <= -3.4)
		tmp = t_1;
	elseif (l <= 3.3e-5)
		tmp = U + (l * (t_0 * (J * 2.0)));
	elseif (l <= 7e+52)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 5.0], $MachinePrecision] * N[(t$95$0 * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4], t$95$1, If[LessEqual[l, 3.3e-5], N[(U + N[(l * N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7e+52], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.39999999999999991 or 7e52 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around inf 95.9%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 95.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 95.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.016666666666666666 \cdot J\right)} + U \]

      3. *-commutative 95.0%

         \[\leadsto \left({\ell}^{5} \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \left(0.016666666666666666 \cdot J\right) + U \]

      4. associate-*l* 95.0%

         \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.016666666666666666 \cdot J\right)\right)} + U \]

      5. *-commutative 95.0%

         \[\leadsto {\ell}^{5} \cdot \color{blue}{\left(\left(0.016666666666666666 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      6. *-commutative 95.0%

         \[\leadsto {\ell}^{5} \cdot \left(\color{blue}{\left(J \cdot 0.016666666666666666\right)} \cdot \cos \left(K \cdot 0.5\right)\right) + U \]

      7. *-commutative 95.0%

         \[\leadsto {\ell}^{5} \cdot \left(\left(J \cdot 0.016666666666666666\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   6. Simplified 95.0%

      \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(\left(J \cdot 0.016666666666666666\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -3.39999999999999991 < l < 3.3000000000000003e-5

   1. Initial program 70.3%

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

   3. Taylor expanded in l around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

   5. Simplified 99.8%

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

   if 3.3000000000000003e-5 < l < 7e52

   1. Initial program 98.7%

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

   3. Taylor expanded in K around 0 84.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+52}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{5} \cdot \left(t\_0 \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.0062:\\ \;\;\;\;U + J \cdot \left(t\_0 \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 9.4 \cdot 10^{+52}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 5.0) (* t_0 (* J 0.016666666666666666))))))
   (if (<= l -5.0)
     t_1
     (if (<= l 0.0062)
       (+ U (* J (* t_0 (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
       (if (<= l 9.4e+52) (+ (* (- (exp l) (exp (- l))) J) U) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -5.0) {
		tmp = t_1;
	} else if (l <= 0.0062) {
		tmp = U + (J * (t_0 * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 9.4e+52) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((l ** 5.0d0) * (t_0 * (j * 0.016666666666666666d0)))
    if (l <= (-5.0d0)) then
        tmp = t_1
    else if (l <= 0.0062d0) then
        tmp = u + (j * (t_0 * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 9.4d+52) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)));
	double tmp;
	if (l <= -5.0) {
		tmp = t_1;
	} else if (l <= 0.0062) {
		tmp = U + (J * (t_0 * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 9.4e+52) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)))
	tmp = 0
	if l <= -5.0:
		tmp = t_1
	elif l <= 0.0062:
		tmp = U + (J * (t_0 * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 9.4e+52:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 5.0) * Float64(t_0 * Float64(J * 0.016666666666666666))))
	tmp = 0.0
	if (l <= -5.0)
		tmp = t_1;
	elseif (l <= 0.0062)
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 9.4e+52)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((l ^ 5.0) * (t_0 * (J * 0.016666666666666666)));
	tmp = 0.0;
	if (l <= -5.0)
		tmp = t_1;
	elseif (l <= 0.0062)
		tmp = U + (J * (t_0 * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 9.4e+52)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 5.0], $MachinePrecision] * N[(t$95$0 * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.0], t$95$1, If[LessEqual[l, 0.0062], N[(U + N[(J * N[(t$95$0 * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.4e+52], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5 or 9.3999999999999999e52 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around inf 95.9%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 95.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 95.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.016666666666666666 \cdot J\right)} + U \]

      3. *-commutative 95.0%

         \[\leadsto \left({\ell}^{5} \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \left(0.016666666666666666 \cdot J\right) + U \]

      4. associate-*l* 95.0%

         \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.016666666666666666 \cdot J\right)\right)} + U \]

      5. *-commutative 95.0%

         \[\leadsto {\ell}^{5} \cdot \color{blue}{\left(\left(0.016666666666666666 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      6. *-commutative 95.0%

         \[\leadsto {\ell}^{5} \cdot \left(\color{blue}{\left(J \cdot 0.016666666666666666\right)} \cdot \cos \left(K \cdot 0.5\right)\right) + U \]

      7. *-commutative 95.0%

         \[\leadsto {\ell}^{5} \cdot \left(\left(J \cdot 0.016666666666666666\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   6. Simplified 95.0%

      \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(\left(J \cdot 0.016666666666666666\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -5 < l < 0.00619999999999999978

   1. Initial program 70.6%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. associate-*r* 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*r* 99.9%

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

      10. distribute-rgt-out 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*r* 99.9%

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

      14. *-commutative 99.9%

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

      15. distribute-rgt-in 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in J around 0 99.9%

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

   if 0.00619999999999999978 < l < 9.3999999999999999e52

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0 83.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq 0.0062:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 9.4 \cdot 10^{+52}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.046:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.046)
     (+ U (* t_0 (* l (* J 2.0))))
     (+ U (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.046) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.046d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u + ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.046) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.046:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.046)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.046)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.046], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.045999999999999999

   1. Initial program 82.0%

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

   3. Taylor expanded in l around 0 76.0%

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

      1. associate-*r* 40.8%

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

      2. *-commutative 40.8%

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

   5. Simplified 76.0%

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

   if 0.045999999999999999 < (cos.f64 (/.f64 K 2))

   1. Initial program 82.7%

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

   3. Taylor expanded in K around 0 82.7%

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

   4. Taylor expanded in l around 0 85.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.046:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.046:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.046)
     (+ U (* t_0 (* l (* J 2.0))))
     (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.046) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.046d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.046) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.046:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.046)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.046)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.046], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.045999999999999999

   1. Initial program 82.0%

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

   3. Taylor expanded in l around 0 76.0%

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

      1. associate-*r* 40.8%

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

      2. *-commutative 40.8%

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

   5. Simplified 76.0%

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

   if 0.045999999999999999 < (cos.f64 (/.f64 K 2))

   1. Initial program 82.7%

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

   3. Taylor expanded in l around 0 94.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in l around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-*r* 89.3%

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

      3. *-commutative 89.3%

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

      4. associate-*r* 89.3%

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

      5. *-commutative 89.3%

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

      6. associate-*r* 89.3%

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

      7. associate-*r* 89.3%

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

      8. *-commutative 89.3%

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

      9. associate-*r* 89.3%

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

      10. distribute-rgt-out 89.3%

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

      11. +-commutative 89.3%

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

      12. *-commutative 89.3%

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

      13. associate-*r* 89.3%

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

      14. *-commutative 89.3%

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

      15. distribute-rgt-in 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in K around 0 85.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.046:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -380:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -380.0)
     t_0
     (if (<= l 3.3e-5) (+ U (* l (* (cos (* K 0.5)) (* J 2.0)))) (+ t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -380.0) {
		tmp = t_0;
	} else if (l <= 3.3e-5) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    if (l <= (-380.0d0)) then
        tmp = t_0
    else if (l <= 3.3d-5) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = t_0 + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -380.0) {
		tmp = t_0;
	} else if (l <= 3.3e-5) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -380.0:
		tmp = t_0
	elif l <= 3.3e-5:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = t_0 + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -380.0)
		tmp = t_0;
	elseif (l <= 3.3e-5)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = Float64(t_0 + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -380.0)
		tmp = t_0;
	elseif (l <= 3.3e-5)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = t_0 + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -380.0], t$95$0, If[LessEqual[l, 3.3e-5], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -380

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 71.2%

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

   4. Taylor expanded in J around inf 71.2%

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

   if -380 < l < 3.3000000000000003e-5

   1. Initial program 70.3%

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

   3. Taylor expanded in l around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

   5. Simplified 99.8%

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

   if 3.3000000000000003e-5 < l

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0 79.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -380:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in l around 0 75.6%

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

4. Final simplification 75.6%

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

Alternative 9: 63.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in l around 0 75.6%

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

   1. associate-*r* 75.6%

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

   2. *-commutative 75.6%

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

   3. associate-*l* 75.6%

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

5. Simplified 75.6%

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

6. Final simplification 75.6%

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

Alternative 10: 41.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-18} \lor \neg \left(\ell \leq 520\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7.2e-18) (not (<= l 520.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.2e-18) || !(l <= 520.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-7.2d-18)) .or. (.not. (l <= 520.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.2e-18) || !(l <= 520.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7.2e-18) or not (l <= 520.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7.2e-18) || !(l <= 520.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -7.2e-18) || ~((l <= 520.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7.2e-18], N[Not[LessEqual[l, 520.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -7.20000000000000021e-18 or 520 < l

   1. Initial program 99.0%

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

   4. Applied egg-rr 16.9%

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

   if -7.20000000000000021e-18 < l < 520

   1. Initial program 71.4%

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

   3. Taylor expanded in J around 0 69.6%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-18} \lor \neg \left(\ell \leq 520\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 82.5%

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

3. Taylor expanded in K around 0 72.1%

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

4. Taylor expanded in l around 0 63.9%

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

   1. associate-*r* 63.9%

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

   2. *-commutative 63.9%

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

6. Simplified 63.9%

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

7. Final simplification 63.9%

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

Alternative 12: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -0.3333333333333333)

C

double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

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 = -0.3333333333333333d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

Python

def code(J, l, K, U):
	return -0.3333333333333333

Julia

function code(J, l, K, U)
	return -0.3333333333333333
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -0.3333333333333333;
end

Wolfram

code[J_, l_, K_, U_] := -0.3333333333333333

Derivation

1. Initial program 82.5%

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

4. Applied egg-rr 2.4%

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

   1. associate-+r+ 2.4%

      \[\leadsto \frac{U}{\color{blue}{\left(U + -4 \cdot U\right) + U \cdot \left(-4 \cdot U\right)}} \]

   2. distribute-rgt1-in 2.4%

      \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

   3. metadata-eval 2.4%

      \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

   4. *-commutative 2.4%

      \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

   5. distribute-lft-out 2.4%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

   6. associate-/r* 2.4%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

   7. +-commutative 2.4%

      \[\leadsto \frac{\frac{U}{U}}{\color{blue}{-4 \cdot U + -3}} \]

   8. *-inverses 2.4%

      \[\leadsto \frac{\color{blue}{1}}{-4 \cdot U + -3} \]

   9. *-commutative 2.4%

      \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

6. Simplified 2.4%

   \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

7. Taylor expanded in U around 0 2.8%

   \[\leadsto \color{blue}{-0.3333333333333333} \]

8. Final simplification 2.8%

   \[\leadsto -0.3333333333333333 \]
9. Add Preprocessing

Alternative 13: 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 82.5%

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

4. Applied egg-rr 3.0%

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

   1. *-inverses 3.0%

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

6. Simplified 3.0%

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

7. Final simplification 3.0%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 14: 36.1% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in J around 0 42.6%

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

4. Final simplification 42.6%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))