Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.3%

Time: 16.5s

Alternatives: 17

Speedup: 1.3×

• 50.9% 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: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot \left(J \cdot 2\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (log1p (expm1 (* (cos (* 0.5 K)) l))) (* J 2.0)) U))

C

double code(double J, double l, double K, double U) {
	return (log1p(expm1((cos((0.5 * K)) * l))) * (J * 2.0)) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (Math.log1p(Math.expm1((Math.cos((0.5 * K)) * l))) * (J * 2.0)) + U;
}

Python

def code(J, l, K, U):
	return (math.log1p(math.expm1((math.cos((0.5 * K)) * l))) * (J * 2.0)) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(log1p(expm1(Float64(cos(Float64(0.5 * K)) * l))) * Float64(J * 2.0)) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[Log[1 + N[(Exp[N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] + U), $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. Add Preprocessing

3. Taylor expanded in l around 0 61.8%

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

   1. associate-*r* 61.8%

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

5. Simplified 61.8%

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

6. Taylor expanded in J around 0 61.8%

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

   1. associate-*r* 61.8%

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

   2. *-commutative 61.8%

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

   3. *-commutative 61.8%

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

   4. *-commutative 61.8%

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

8. Simplified 61.8%

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

   1. log1p-expm1-u 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

10. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]
11. Add Preprocessing

Alternative 2: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.82:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{t\_0 \cdot \ell}{U}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.55:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.005:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(t\_0 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.82)
     (* U (+ 1.0 (* 2.0 (* J (/ (* t_0 l) U)))))
     (if (<= t_1 -0.55)
       (+ U (* (* J 2.0) (+ l (* l (* -0.125 (pow K 2.0))))))
       (if (<= t_1 -0.005)
         (+ U (* 2.0 (* l (* t_0 J))))
         (+ U (* (* J 2.0) (log1p (expm1 l)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.82) {
		tmp = U * (1.0 + (2.0 * (J * ((t_0 * l) / U))));
	} else if (t_1 <= -0.55) {
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * pow(K, 2.0)))));
	} else if (t_1 <= -0.005) {
		tmp = U + (2.0 * (l * (t_0 * J)));
	} else {
		tmp = U + ((J * 2.0) * log1p(expm1(l)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.82) {
		tmp = U * (1.0 + (2.0 * (J * ((t_0 * l) / U))));
	} else if (t_1 <= -0.55) {
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * Math.pow(K, 2.0)))));
	} else if (t_1 <= -0.005) {
		tmp = U + (2.0 * (l * (t_0 * J)));
	} else {
		tmp = U + ((J * 2.0) * Math.log1p(Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -0.82:
		tmp = U * (1.0 + (2.0 * (J * ((t_0 * l) / U))))
	elif t_1 <= -0.55:
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * math.pow(K, 2.0)))))
	elif t_1 <= -0.005:
		tmp = U + (2.0 * (l * (t_0 * J)))
	else:
		tmp = U + ((J * 2.0) * math.log1p(math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.82)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(t_0 * l) / U)))));
	elseif (t_1 <= -0.55)
		tmp = Float64(U + Float64(Float64(J * 2.0) * Float64(l + Float64(l * Float64(-0.125 * (K ^ 2.0))))));
	elseif (t_1 <= -0.005)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(t_0 * J))));
	else
		tmp = Float64(U + Float64(Float64(J * 2.0) * log1p(expm1(l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.82], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(t$95$0 * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.55], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[(l + N[(l * N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.005], N[(U + N[(2.0 * N[(l * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951

   1. Initial program 73.4%

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

   3. Taylor expanded in l around 0 62.7%

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

      1. associate-*r* 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in U around inf 62.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 75.7%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

      2. *-commutative 75.7%

         \[\leadsto U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\color{blue}{\cos \left(0.5 \cdot K\right) \cdot \ell}}{U}\right)\right) \]

   8. Simplified 75.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \ell}{U}\right)\right)} \]

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.55000000000000004

   1. Initial program 97.5%

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

   3. Taylor expanded in l around 0 29.0%

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

      1. associate-*r* 29.0%

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

   5. Simplified 29.0%

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

   6. Taylor expanded in J around 0 29.0%

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

      1. associate-*r* 29.0%

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

      2. *-commutative 29.0%

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

      3. *-commutative 29.0%

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

      4. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in K around 0 70.2%

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

       1. associate-*r* 70.2%

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

   11. Simplified 70.2%

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

   if -0.55000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 65.8%

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

   3. Taylor expanded in l around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in J around 0 72.2%

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

      1. associate-*r* 72.2%

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

      2. *-commutative 72.2%

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

      3. *-commutative 72.2%

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

      4. *-commutative 72.2%

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

   8. Simplified 72.2%

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

      1. log1p-expm1-u 99.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around inf 72.2%

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

       1. *-commutative 72.2%

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

       2. associate-*l* 72.2%

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

       3. *-commutative 72.2%

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

   13. Simplified 72.2%

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

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

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0 63.9%

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

      1. associate-*r* 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in J around 0 63.9%

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

      1. associate-*r* 63.9%

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

      2. *-commutative 63.9%

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

      3. *-commutative 63.9%

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

      4. *-commutative 63.9%

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

   8. Simplified 63.9%

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

      1. log1p-expm1-u 99.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 94.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell}\right)\right) \cdot \left(J \cdot 2\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \ell}{U}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.55:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \ell}{U}\right)\right)\\ \mathbf{if}\;t\_0 \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.55:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* U (+ 1.0 (* 2.0 (* J (/ (* (cos (* 0.5 K)) l) U)))))))
   (if (<= t_0 -0.82)
     t_1
     (if (<= t_0 -0.55)
       (+ U (* (* J 2.0) (+ l (* l (* -0.125 (pow K 2.0))))))
       (if (<= t_0 0.2)
         t_1
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U * (1.0 + (2.0 * (J * ((cos((0.5 * K)) * l) / U))));
	double tmp;
	if (t_0 <= -0.82) {
		tmp = t_1;
	} else if (t_0 <= -0.55) {
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * pow(K, 2.0)))));
	} else if (t_0 <= 0.2) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u * (1.0d0 + (2.0d0 * (j * ((cos((0.5d0 * k)) * l) / u))))
    if (t_0 <= (-0.82d0)) then
        tmp = t_1
    else if (t_0 <= (-0.55d0)) then
        tmp = u + ((j * 2.0d0) * (l + (l * ((-0.125d0) * (k ** 2.0d0)))))
    else if (t_0 <= 0.2d0) then
        tmp = t_1
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U * (1.0 + (2.0 * (J * ((Math.cos((0.5 * K)) * l) / U))));
	double tmp;
	if (t_0 <= -0.82) {
		tmp = t_1;
	} else if (t_0 <= -0.55) {
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * Math.pow(K, 2.0)))));
	} else if (t_0 <= 0.2) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U * (1.0 + (2.0 * (J * ((math.cos((0.5 * K)) * l) / U))))
	tmp = 0
	if t_0 <= -0.82:
		tmp = t_1
	elif t_0 <= -0.55:
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * math.pow(K, 2.0)))))
	elif t_0 <= 0.2:
		tmp = t_1
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(cos(Float64(0.5 * K)) * l) / U)))))
	tmp = 0.0
	if (t_0 <= -0.82)
		tmp = t_1;
	elseif (t_0 <= -0.55)
		tmp = Float64(U + Float64(Float64(J * 2.0) * Float64(l + Float64(l * Float64(-0.125 * (K ^ 2.0))))));
	elseif (t_0 <= 0.2)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U * (1.0 + (2.0 * (J * ((cos((0.5 * K)) * l) / U))));
	tmp = 0.0;
	if (t_0 <= -0.82)
		tmp = t_1;
	elseif (t_0 <= -0.55)
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * (K ^ 2.0)))));
	elseif (t_0 <= 0.2)
		tmp = t_1;
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.82], t$95$1, If[LessEqual[t$95$0, -0.55], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[(l + N[(l * N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.2], t$95$1, N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.819999999999999951 or -0.55000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 71.2%

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

   3. Taylor expanded in l around 0 65.6%

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

      1. associate-*r* 65.6%

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

   5. Simplified 65.6%

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

   6. Taylor expanded in U around inf 65.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 74.3%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

      2. *-commutative 74.3%

         \[\leadsto U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\color{blue}{\cos \left(0.5 \cdot K\right) \cdot \ell}}{U}\right)\right) \]

   8. Simplified 74.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \ell}{U}\right)\right)} \]

   if -0.819999999999999951 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.55000000000000004

   1. Initial program 97.5%

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

   3. Taylor expanded in l around 0 29.0%

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

      1. associate-*r* 29.0%

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

   5. Simplified 29.0%

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

   6. Taylor expanded in J around 0 29.0%

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

      1. associate-*r* 29.0%

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

      2. *-commutative 29.0%

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

      3. *-commutative 29.0%

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

      4. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in K around 0 70.2%

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

       1. associate-*r* 70.2%

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

   11. Simplified 70.2%

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

   if 0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0 80.3%

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

   4. Taylor expanded in K around 0 76.4%

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

   5. Taylor expanded in J around 0 80.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \ell}{U}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.55:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.2:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.55:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.4:\\ \;\;\;\;U + 2 \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.55)
     (+ U (* (* J 2.0) (+ l (* l (* -0.125 (pow K 2.0))))))
     (if (<= t_0 0.4)
       (+ U (* 2.0 (* (* (cos (* 0.5 K)) l) J)))
       (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.55) {
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * pow(K, 2.0)))));
	} else if (t_0 <= 0.4) {
		tmp = U + (2.0 * ((cos((0.5 * K)) * l) * J));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(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 <= (-0.55d0)) then
        tmp = u + ((j * 2.0d0) * (l + (l * ((-0.125d0) * (k ** 2.0d0)))))
    else if (t_0 <= 0.4d0) then
        tmp = u + (2.0d0 * ((cos((0.5d0 * k)) * l) * j))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (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 <= -0.55) {
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * Math.pow(K, 2.0)))));
	} else if (t_0 <= 0.4) {
		tmp = U + (2.0 * ((Math.cos((0.5 * K)) * l) * J));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.55:
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * math.pow(K, 2.0)))))
	elif t_0 <= 0.4:
		tmp = U + (2.0 * ((math.cos((0.5 * K)) * l) * J))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.55)
		tmp = Float64(U + Float64(Float64(J * 2.0) * Float64(l + Float64(l * Float64(-0.125 * (K ^ 2.0))))));
	elseif (t_0 <= 0.4)
		tmp = Float64(U + Float64(2.0 * Float64(Float64(cos(Float64(0.5 * K)) * l) * J)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.55)
		tmp = U + ((J * 2.0) * (l + (l * (-0.125 * (K ^ 2.0)))));
	elseif (t_0 <= 0.4)
		tmp = U + (2.0 * ((cos((0.5 * K)) * l) * J));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.55], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[(l + N[(l * N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.4], N[(U + N[(2.0 * N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.55000000000000004

   1. Initial program 84.6%

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

   3. Taylor expanded in l around 0 47.1%

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

      1. associate-*r* 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in J around 0 47.0%

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

      1. associate-*r* 47.0%

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

      2. *-commutative 47.0%

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

      3. *-commutative 47.0%

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

      4. *-commutative 47.0%

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

   8. Simplified 47.0%

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

   9. Taylor expanded in K around 0 58.3%

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

       1. associate-*r* 58.3%

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

   11. Simplified 58.3%

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

   if -0.55000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.40000000000000002

   1. Initial program 66.8%

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

   3. Taylor expanded in l around 0 70.0%

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

   if 0.40000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0 79.6%

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

   4. Taylor expanded in K around 0 77.0%

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

   5. Taylor expanded in J around 0 81.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.55:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.4:\\ \;\;\;\;U + 2 \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.5)
     (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
     (+ U (* (* J 2.0) (log1p (expm1 l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.5) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else {
		tmp = U + ((J * 2.0) * log1p(expm1(l)));
	}
	return tmp;
}

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 <= 0.5) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else {
		tmp = U + ((J * 2.0) * Math.log1p(Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.5:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	else:
		tmp = U + ((J * 2.0) * math.log1p(math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	else
		tmp = Float64(U + Float64(Float64(J * 2.0) * log1p(expm1(l))));
	end
	return 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, 0.5], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 76.5%

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

   3. Taylor expanded in l around 0 88.2%

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

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.7%

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

   3. Taylor expanded in l around 0 64.1%

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

      1. associate-*r* 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in J around 0 64.1%

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

      1. associate-*r* 64.1%

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

      2. *-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. *-commutative 64.1%

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

   8. Simplified 64.1%

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

      1. log1p-expm1-u 99.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 97.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell}\right)\right) \cdot \left(J \cdot 2\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.5)
   (+
    U
    (* J (* l (* (cos (* 0.5 K)) (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
   (+ U (* (* J 2.0) (log1p (expm1 l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.5) {
		tmp = U + (J * (l * (cos((0.5 * K)) * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else {
		tmp = U + ((J * 2.0) * log1p(expm1(l)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.5) {
		tmp = U + (J * (l * (Math.cos((0.5 * K)) * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else {
		tmp = U + ((J * 2.0) * Math.log1p(Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.5:
		tmp = U + (J * (l * (math.cos((0.5 * K)) * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	else:
		tmp = U + ((J * 2.0) * math.log1p(math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.5)
		tmp = Float64(U + Float64(J * Float64(l * Float64(cos(Float64(0.5 * K)) * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	else
		tmp = Float64(U + Float64(Float64(J * 2.0) * log1p(expm1(l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.5], N[(U + N[(J * N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 76.5%

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

   3. Taylor expanded in l around 0 83.8%

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

   4. Taylor expanded in J around 0 88.1%

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

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.7%

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

   3. Taylor expanded in l around 0 64.1%

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

      1. associate-*r* 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in J around 0 64.1%

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

      1. associate-*r* 64.1%

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

      2. *-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. *-commutative 64.1%

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

   8. Simplified 64.1%

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

      1. log1p-expm1-u 99.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 97.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell}\right)\right) \cdot \left(J \cdot 2\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -340:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* 0.3333333333333333 (* J (pow l 3.0)))))))
   (if (<= l -2.6e+119)
     t_0
     (if (<= l -340.0)
       (+ U (* (* J 2.0) (log1p (expm1 l))))
       (if (<= l 5.9e-5)
         (+ U (* 2.0 (* l (* (cos (* 0.5 K)) J))))
         (if (<= l 7.8e+99) (+ U (* J (- (exp l) (exp (- l))))) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (0.3333333333333333 * (J * pow(l, 3.0))));
	double tmp;
	if (l <= -2.6e+119) {
		tmp = t_0;
	} else if (l <= -340.0) {
		tmp = U + ((J * 2.0) * log1p(expm1(l)));
	} else if (l <= 5.9e-5) {
		tmp = U + (2.0 * (l * (cos((0.5 * K)) * J)));
	} else if (l <= 7.8e+99) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (0.3333333333333333 * (J * Math.pow(l, 3.0))));
	double tmp;
	if (l <= -2.6e+119) {
		tmp = t_0;
	} else if (l <= -340.0) {
		tmp = U + ((J * 2.0) * Math.log1p(Math.expm1(l)));
	} else if (l <= 5.9e-5) {
		tmp = U + (2.0 * (l * (Math.cos((0.5 * K)) * J)));
	} else if (l <= 7.8e+99) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (0.3333333333333333 * (J * math.pow(l, 3.0))))
	tmp = 0
	if l <= -2.6e+119:
		tmp = t_0
	elif l <= -340.0:
		tmp = U + ((J * 2.0) * math.log1p(math.expm1(l)))
	elif l <= 5.9e-5:
		tmp = U + (2.0 * (l * (math.cos((0.5 * K)) * J)))
	elif l <= 7.8e+99:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))))
	tmp = 0.0
	if (l <= -2.6e+119)
		tmp = t_0;
	elseif (l <= -340.0)
		tmp = Float64(U + Float64(Float64(J * 2.0) * log1p(expm1(l))));
	elseif (l <= 5.9e-5)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(cos(Float64(0.5 * K)) * J))));
	elseif (l <= 7.8e+99)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.6e+119], t$95$0, If[LessEqual[l, -340.0], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.9e-5], N[(U + N[(2.0 * N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+99], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.6e119 or 7.79999999999999989e99 < 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 89.5%

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

   4. Taylor expanded in l around inf 100.0%

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

   if -2.6e119 < l < -340

   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 12.4%

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

      1. associate-*r* 12.4%

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

   5. Simplified 12.4%

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

   6. Taylor expanded in J around 0 12.4%

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

      1. associate-*r* 12.4%

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

      2. *-commutative 12.4%

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

      3. *-commutative 12.4%

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

      4. *-commutative 12.4%

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

   8. Simplified 12.4%

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

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 81.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\ell}\right)\right) \cdot \left(J \cdot 2\right) + U \]

   if -340 < l < 5.8999999999999998e-5

   1. Initial program 66.0%

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

   3. Taylor expanded in l around 0 99.4%

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

      1. associate-*r* 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in J around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. *-commutative 99.4%

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

   8. Simplified 99.4%

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

      1. log1p-expm1-u 99.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around inf 99.4%

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

       1. *-commutative 99.4%

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

       2. associate-*l* 99.4%

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

       3. *-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 5.8999999999999998e-5 < l < 7.79999999999999989e99

   1. Initial program 98.9%

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

   3. Taylor expanded in K around 0 82.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+119}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -340:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.3e+42)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 1.02e-5)
     (+ U (* 2.0 (* l (* (cos (* 0.5 K)) J))))
     (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.3e+42) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 1.02e-5) {
		tmp = U + (2.0 * (l * (cos((0.5 * K)) * J)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(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 <= (-5.3d+42)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 1.02d-5) then
        tmp = u + (2.0d0 * (l * (cos((0.5d0 * k)) * j)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (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 <= -5.3e+42) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 1.02e-5) {
		tmp = U + (2.0 * (l * (Math.cos((0.5 * K)) * J)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.3e+42:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 1.02e-5:
		tmp = U + (2.0 * (l * (math.cos((0.5 * K)) * J)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.3e+42)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 1.02e-5)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(cos(Float64(0.5 * K)) * J))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.3e+42)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 1.02e-5)
		tmp = U + (2.0 * (l * (cos((0.5 * K)) * J)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.3e+42], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.02e-5], N[(U + N[(2.0 * N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.30000000000000028e42

   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 72.5%

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

   4. Taylor expanded in K around 0 56.6%

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

   5. Taylor expanded in l around inf 66.4%

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

   if -5.30000000000000028e42 < l < 1.0200000000000001e-5

   1. Initial program 69.1%

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

   3. Taylor expanded in l around 0 91.5%

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

      1. associate-*r* 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in J around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. *-commutative 91.5%

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

      3. *-commutative 91.5%

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

      4. *-commutative 91.5%

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

   8. Simplified 91.5%

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

      1. log1p-expm1-u 99.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around inf 91.5%

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

       1. *-commutative 91.5%

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

       2. associate-*l* 91.5%

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

       3. *-commutative 91.5%

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

   13. Simplified 91.5%

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

   if 1.0200000000000001e-5 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 65.5%

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

   4. Taylor expanded in K around 0 48.1%

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

   5. Taylor expanded in J around 0 50.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 0.00027:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.4e+42)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 0.00027)
     (+ U (* 2.0 (* l (* (cos (* 0.5 K)) J))))
     (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.4e+42) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 0.00027) {
		tmp = U + (2.0 * (l * (cos((0.5 * K)) * J)));
	} else {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(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 <= (-5.4d+42)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 0.00027d0) then
        tmp = u + (2.0d0 * (l * (cos((0.5d0 * k)) * j)))
    else
        tmp = j * (l * (2.0d0 + (0.3333333333333333d0 * (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 <= -5.4e+42) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 0.00027) {
		tmp = U + (2.0 * (l * (Math.cos((0.5 * K)) * J)));
	} else {
		tmp = J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.4e+42:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 0.00027:
		tmp = U + (2.0 * (l * (math.cos((0.5 * K)) * J)))
	else:
		tmp = J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.4e+42)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 0.00027)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(cos(Float64(0.5 * K)) * J))));
	else
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.4e+42)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 0.00027)
		tmp = U + (2.0 * (l * (cos((0.5 * K)) * J)));
	else
		tmp = J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.4e+42], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00027], N[(U + N[(2.0 * N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.4000000000000001e42

   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 72.5%

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

   4. Taylor expanded in K around 0 56.6%

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

   5. Taylor expanded in l around inf 66.4%

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

   if -5.4000000000000001e42 < l < 2.70000000000000003e-4

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 91.5%

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

      1. associate-*r* 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in J around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. *-commutative 91.5%

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

      3. *-commutative 91.5%

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

      4. *-commutative 91.5%

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

   8. Simplified 91.5%

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

      1. log1p-expm1-u 99.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around inf 91.5%

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

       1. *-commutative 91.5%

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

       2. associate-*l* 91.6%

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

       3. *-commutative 91.6%

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

   13. Simplified 91.6%

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

   if 2.70000000000000003e-4 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 64.9%

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

   4. Taylor expanded in K around 0 47.3%

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

   5. Taylor expanded in J around inf 49.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 0.00027:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 0.00027:\\ \;\;\;\;U + 2 \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.3e+42)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 0.00027)
     (+ U (* 2.0 (* (* (cos (* 0.5 K)) l) J)))
     (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.3e+42) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 0.00027) {
		tmp = U + (2.0 * ((cos((0.5 * K)) * l) * J));
	} else {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(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 <= (-5.3d+42)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 0.00027d0) then
        tmp = u + (2.0d0 * ((cos((0.5d0 * k)) * l) * j))
    else
        tmp = j * (l * (2.0d0 + (0.3333333333333333d0 * (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 <= -5.3e+42) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 0.00027) {
		tmp = U + (2.0 * ((Math.cos((0.5 * K)) * l) * J));
	} else {
		tmp = J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.3e+42:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 0.00027:
		tmp = U + (2.0 * ((math.cos((0.5 * K)) * l) * J))
	else:
		tmp = J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.3e+42)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 0.00027)
		tmp = Float64(U + Float64(2.0 * Float64(Float64(cos(Float64(0.5 * K)) * l) * J)));
	else
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.3e+42)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 0.00027)
		tmp = U + (2.0 * ((cos((0.5 * K)) * l) * J));
	else
		tmp = J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.3e+42], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00027], N[(U + N[(2.0 * N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.30000000000000028e42

   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 72.5%

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

   4. Taylor expanded in K around 0 56.6%

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

   5. Taylor expanded in l around inf 66.4%

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

   if -5.30000000000000028e42 < l < 2.70000000000000003e-4

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 91.5%

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

   if 2.70000000000000003e-4 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 64.9%

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

   4. Taylor expanded in K around 0 47.3%

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

   5. Taylor expanded in J around inf 49.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 0.00027:\\ \;\;\;\;U + 2 \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+26}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 0.00027:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.2e+26)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 0.00027)
     (fma l (* J 2.0) U)
     (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.2e+26) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 0.00027) {
		tmp = fma(l, (J * 2.0), U);
	} else {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.2e+26)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 0.00027)
		tmp = fma(l, Float64(J * 2.0), U);
	else
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.2e+26], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00027], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.20000000000000007e26

   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 67.6%

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

   4. Taylor expanded in K around 0 51.5%

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

   5. Taylor expanded in l around inf 60.4%

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

   if -2.20000000000000007e26 < l < 2.70000000000000003e-4

   1. Initial program 67.9%

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

   3. Taylor expanded in l around 0 95.0%

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

      1. associate-*r* 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in J around 0 94.9%

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

      1. associate-*r* 94.9%

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

      2. *-commutative 94.9%

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

      3. *-commutative 94.9%

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

      4. *-commutative 94.9%

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

   8. Simplified 94.9%

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

      1. log1p-expm1-u 99.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 76.4%

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

       1. +-commutative 76.4%

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

       2. *-commutative 76.4%

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

       3. *-commutative 76.4%

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

       4. associate-*r* 76.4%

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

       5. fma-define 76.4%

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

       6. *-commutative 76.4%

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

   13. Simplified 76.4%

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

   if 2.70000000000000003e-4 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 64.9%

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

   4. Taylor expanded in K around 0 47.3%

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

   5. Taylor expanded in J around inf 49.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+26}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 0.00027:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.3e+26)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 9e+32)
     (fma l (* J 2.0) U)
     (+ U (* (pow l 3.0) (* J 0.3333333333333333))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.3e+26) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 9e+32) {
		tmp = fma(l, (J * 2.0), U);
	} else {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.3e+26)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 9e+32)
		tmp = fma(l, Float64(J * 2.0), U);
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.3e+26], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e+32], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.3000000000000001e26

   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 67.6%

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

   4. Taylor expanded in K around 0 51.5%

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

   5. Taylor expanded in l around inf 60.4%

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

   if -2.3000000000000001e26 < l < 9.0000000000000007e32

   1. Initial program 69.8%

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

   3. Taylor expanded in l around 0 89.8%

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

      1. associate-*r* 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in J around 0 89.7%

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

      1. associate-*r* 89.7%

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

      2. *-commutative 89.7%

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

      3. *-commutative 89.7%

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

      4. *-commutative 89.7%

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

   8. Simplified 89.7%

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

      1. log1p-expm1-u 98.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 72.4%

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

       1. +-commutative 72.4%

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

       2. *-commutative 72.4%

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

       3. *-commutative 72.4%

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

       4. associate-*r* 72.4%

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

       5. fma-define 72.4%

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

       6. *-commutative 72.4%

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

   13. Simplified 72.4%

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

   if 9.0000000000000007e32 < 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 69.6%

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

   4. Taylor expanded in l around inf 78.0%

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

   5. Taylor expanded in K around 0 54.7%

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

      1. *-commutative 54.7%

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

      2. *-commutative 54.7%

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

      3. associate-*l* 54.7%

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

   7. Simplified 54.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+26} \lor \neg \left(\ell \leq 2000000000000\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.2e+26) (not (<= l 2000000000000.0)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (fma l (* J 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.2e+26) || !(l <= 2000000000000.0)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else {
		tmp = fma(l, (J * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.2e+26) || !(l <= 2000000000000.0))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	else
		tmp = fma(l, Float64(J * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.2e+26], N[Not[LessEqual[l, 2000000000000.0]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.20000000000000007e26 or 2e12 < 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 67.7%

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

   4. Taylor expanded in K around 0 50.3%

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

   5. Taylor expanded in l around inf 56.3%

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

   if -2.20000000000000007e26 < l < 2e12

   1. Initial program 69.1%

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

   3. Taylor expanded in l around 0 91.7%

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

      1. associate-*r* 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in J around 0 91.7%

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

      1. associate-*r* 91.7%

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

      2. *-commutative 91.7%

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

      3. *-commutative 91.7%

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

      4. *-commutative 91.7%

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

   8. Simplified 91.7%

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

      1. log1p-expm1-u 98.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   10. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

   11. Taylor expanded in K around 0 74.0%

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

       1. +-commutative 74.0%

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

       2. *-commutative 74.0%

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

       3. *-commutative 74.0%

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

       4. associate-*r* 74.0%

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

       5. fma-define 74.0%

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

       6. *-commutative 74.0%

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

   13. Simplified 74.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+26} \lor \neg \left(\ell \leq 2000000000000\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in l around 0 61.8%

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

   1. associate-*r* 61.8%

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

5. Simplified 61.8%

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

6. Taylor expanded in J around 0 61.8%

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

   1. associate-*r* 61.8%

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

   2. *-commutative 61.8%

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

   3. *-commutative 61.8%

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

   4. *-commutative 61.8%

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

8. Simplified 61.8%

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

   1. log1p-expm1-u 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

10. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} \cdot \left(J \cdot 2\right) + U \]

11. Taylor expanded in K around 0 50.4%

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

    1. +-commutative 50.4%

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

    2. *-commutative 50.4%

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

    3. *-commutative 50.4%

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

    4. associate-*r* 50.4%

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

    5. fma-define 50.4%

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

    6. *-commutative 50.4%

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

13. Simplified 50.4%

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

14. Final simplification 50.4%

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

Alternative 15: 53.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in l around 0 61.8%

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

   1. associate-*r* 61.8%

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

5. Simplified 61.8%

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

6. Taylor expanded in K around 0 50.4%

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

   1. associate-*r* 50.4%

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

   2. *-commutative 50.4%

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

   3. *-commutative 50.4%

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

8. Simplified 50.4%

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

9. Final simplification 50.4%

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

Alternative 16: 39.0% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + \ell \cdot \left(J \cdot -6\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* l (* J -6.0))))

C

double code(double J, double l, double K, double U) {
	return U + (l * (J * -6.0));
}

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 * (-6.0d0)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (l * (J * -6.0));
}

Python

def code(J, l, K, U):
	return U + (l * (J * -6.0))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(l * Float64(J * -6.0)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (l * (J * -6.0));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(l * N[(J * -6.0), $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. Add Preprocessing

3. Taylor expanded in l around 0 61.8%

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

   1. associate-*r* 61.8%

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

5. Simplified 61.8%

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

7. Applied egg-rr 34.7%

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

   1. *-commutative 34.7%

      \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} \cdot -3\right) + U \]

9. Simplified 34.7%

   \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) \cdot -3\right)} + U \]

10. Taylor expanded in K around 0 34.1%

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

    1. *-commutative 34.1%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

    2. *-commutative 34.1%

       \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot -6 + U \]

    3. associate-*l* 34.1%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot -6\right)} + U \]

12. Simplified 34.1%

    \[\leadsto \color{blue}{\ell \cdot \left(J \cdot -6\right)} + U \]

13. Final simplification 34.1%

    \[\leadsto U + \ell \cdot \left(J \cdot -6\right) \]
14. Add Preprocessing

Alternative 17: 36.0% 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. Add Preprocessing

4. Applied egg-rr 25.2%

   \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Taylor expanded in J around 0 32.2%

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

Reproduce

herbie shell --seed 2024088 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))