Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.9% → 99.7%

Time: 11.2s

Alternatives: 17

Speedup: 1.5×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e-5)))
     (+ (* t_0 (* t_1 J)) U)
     (+
      U
      (* t_0 (+ (* 2.0 (* l J)) (* 0.3333333333333333 (* J (pow l 3.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e-5)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * pow(l, 3.0)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e-5)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * Math.pow(l, 3.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e-5):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * math.pow(l, 3.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e-5))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(Float64(2.0 * Float64(l * J)) + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e-5)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * (l ^ 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e-5]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 5.00000000000000024e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.00000000000000024e-5

   1. Initial program 68.5%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 2: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.48:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 13000000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J)))
        (t_1 (* (* (pow l 3.0) (* J 0.3333333333333333)) (cos (* K 0.5)))))
   (if (<= l -1.35e+96)
     t_1
     (if (<= l -0.48)
       t_0
       (if (<= l 13000000000000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (+ (* 2.0 (* l J)) (* 0.3333333333333333 (* J (pow l 3.0))))))
         (if (<= l 1.9e+97) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double t_1 = (pow(l, 3.0) * (J * 0.3333333333333333)) * cos((K * 0.5));
	double tmp;
	if (l <= -1.35e+96) {
		tmp = t_1;
	} else if (l <= -0.48) {
		tmp = t_0;
	} else if (l <= 13000000000000.0) {
		tmp = U + (cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * pow(l, 3.0)))));
	} else if (l <= 1.9e+97) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + ((exp(l) - exp(-l)) * j)
    t_1 = ((l ** 3.0d0) * (j * 0.3333333333333333d0)) * cos((k * 0.5d0))
    if (l <= (-1.35d+96)) then
        tmp = t_1
    else if (l <= (-0.48d0)) then
        tmp = t_0
    else if (l <= 13000000000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * ((2.0d0 * (l * j)) + (0.3333333333333333d0 * (j * (l ** 3.0d0)))))
    else if (l <= 1.9d+97) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double t_1 = (Math.pow(l, 3.0) * (J * 0.3333333333333333)) * Math.cos((K * 0.5));
	double tmp;
	if (l <= -1.35e+96) {
		tmp = t_1;
	} else if (l <= -0.48) {
		tmp = t_0;
	} else if (l <= 13000000000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * Math.pow(l, 3.0)))));
	} else if (l <= 1.9e+97) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.exp(l) - math.exp(-l)) * J)
	t_1 = (math.pow(l, 3.0) * (J * 0.3333333333333333)) * math.cos((K * 0.5))
	tmp = 0
	if l <= -1.35e+96:
		tmp = t_1
	elif l <= -0.48:
		tmp = t_0
	elif l <= 13000000000000.0:
		tmp = U + (math.cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * math.pow(l, 3.0)))))
	elif l <= 1.9e+97:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	t_1 = Float64(Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (l <= -1.35e+96)
		tmp = t_1;
	elseif (l <= -0.48)
		tmp = t_0;
	elseif (l <= 13000000000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(Float64(2.0 * Float64(l * J)) + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))));
	elseif (l <= 1.9e+97)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((exp(l) - exp(-l)) * J);
	t_1 = ((l ^ 3.0) * (J * 0.3333333333333333)) * cos((K * 0.5));
	tmp = 0.0;
	if (l <= -1.35e+96)
		tmp = t_1;
	elseif (l <= -0.48)
		tmp = t_0;
	elseif (l <= 13000000000000.0)
		tmp = U + (cos((K / 2.0)) * ((2.0 * (l * J)) + (0.3333333333333333 * (J * (l ^ 3.0)))));
	elseif (l <= 1.9e+97)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.35e+96], t$95$1, If[LessEqual[l, -0.48], t$95$0, If[LessEqual[l, 13000000000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.9e+97], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.35000000000000011e96 or 1.90000000000000018e97 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 98.9%

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

      1. fma-def 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. associate-*r* 98.9%

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

      5. distribute-rgt-out 98.9%

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

      6. div-inv 98.9%

         \[\leadsto \mathsf{fma}\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right), \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      7. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in l around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. associate-*r* 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-*r* 98.9%

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

      7. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if -1.35000000000000011e96 < l < -0.47999999999999998 or 1.3e13 < l < 1.90000000000000018e97

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 80.0%

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

   if -0.47999999999999998 < l < 1.3e13

   1. Initial program 68.9%

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

   2. Taylor expanded in l around 0 98.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+96}:\\ \;\;\;\;\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -0.48:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 13000000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.056:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 13000000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J)))
        (t_1 (* (* (pow l 3.0) (* J 0.3333333333333333)) (cos (* K 0.5)))))
   (if (<= l -1.35e+96)
     t_1
     (if (<= l -0.056)
       t_0
       (if (<= l 13000000000000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 1.9e+97) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double t_1 = (pow(l, 3.0) * (J * 0.3333333333333333)) * cos((K * 0.5));
	double tmp;
	if (l <= -1.35e+96) {
		tmp = t_1;
	} else if (l <= -0.056) {
		tmp = t_0;
	} else if (l <= 13000000000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.9e+97) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + ((exp(l) - exp(-l)) * j)
    t_1 = ((l ** 3.0d0) * (j * 0.3333333333333333d0)) * cos((k * 0.5d0))
    if (l <= (-1.35d+96)) then
        tmp = t_1
    else if (l <= (-0.056d0)) then
        tmp = t_0
    else if (l <= 13000000000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 1.9d+97) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double t_1 = (Math.pow(l, 3.0) * (J * 0.3333333333333333)) * Math.cos((K * 0.5));
	double tmp;
	if (l <= -1.35e+96) {
		tmp = t_1;
	} else if (l <= -0.056) {
		tmp = t_0;
	} else if (l <= 13000000000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 1.9e+97) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.exp(l) - math.exp(-l)) * J)
	t_1 = (math.pow(l, 3.0) * (J * 0.3333333333333333)) * math.cos((K * 0.5))
	tmp = 0
	if l <= -1.35e+96:
		tmp = t_1
	elif l <= -0.056:
		tmp = t_0
	elif l <= 13000000000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 1.9e+97:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	t_1 = Float64(Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (l <= -1.35e+96)
		tmp = t_1;
	elseif (l <= -0.056)
		tmp = t_0;
	elseif (l <= 13000000000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 1.9e+97)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((exp(l) - exp(-l)) * J);
	t_1 = ((l ^ 3.0) * (J * 0.3333333333333333)) * cos((K * 0.5));
	tmp = 0.0;
	if (l <= -1.35e+96)
		tmp = t_1;
	elseif (l <= -0.056)
		tmp = t_0;
	elseif (l <= 13000000000000.0)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 1.9e+97)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.35e+96], t$95$1, If[LessEqual[l, -0.056], t$95$0, If[LessEqual[l, 13000000000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.9e+97], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.35000000000000011e96 or 1.90000000000000018e97 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 98.9%

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

      1. fma-def 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. associate-*r* 98.9%

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

      5. distribute-rgt-out 98.9%

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

      6. div-inv 98.9%

         \[\leadsto \mathsf{fma}\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right), \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      7. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in l around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. associate-*r* 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-*r* 98.9%

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

      7. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if -1.35000000000000011e96 < l < -0.0560000000000000012 or 1.3e13 < l < 1.90000000000000018e97

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 80.0%

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

   if -0.0560000000000000012 < l < 1.3e13

   1. Initial program 68.9%

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

   2. Taylor expanded in l around 0 98.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+96}:\\ \;\;\;\;\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -0.056:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 13000000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot t_1\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -0.03:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 13000000000000:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(J \cdot 2\right), t_1, U\right)\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J)))
        (t_1 (cos (* K 0.5)))
        (t_2 (* (* (pow l 3.0) (* J 0.3333333333333333)) t_1)))
   (if (<= l -2.6e+94)
     t_2
     (if (<= l -0.03)
       t_0
       (if (<= l 13000000000000.0)
         (fma (* l (* J 2.0)) t_1 U)
         (if (<= l 2.02e+96) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double t_1 = cos((K * 0.5));
	double t_2 = (pow(l, 3.0) * (J * 0.3333333333333333)) * t_1;
	double tmp;
	if (l <= -2.6e+94) {
		tmp = t_2;
	} else if (l <= -0.03) {
		tmp = t_0;
	} else if (l <= 13000000000000.0) {
		tmp = fma((l * (J * 2.0)), t_1, U);
	} else if (l <= 2.02e+96) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	t_1 = cos(Float64(K * 0.5))
	t_2 = Float64(Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)) * t_1)
	tmp = 0.0
	if (l <= -2.6e+94)
		tmp = t_2;
	elseif (l <= -0.03)
		tmp = t_0;
	elseif (l <= 13000000000000.0)
		tmp = fma(Float64(l * Float64(J * 2.0)), t_1, U);
	elseif (l <= 2.02e+96)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[l, -2.6e+94], t$95$2, If[LessEqual[l, -0.03], t$95$0, If[LessEqual[l, 13000000000000.0], N[(N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + U), $MachinePrecision], If[LessEqual[l, 2.02e+96], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.5999999999999999e94 or 2.02000000000000006e96 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 98.9%

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

      1. fma-def 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. associate-*r* 98.9%

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

      5. distribute-rgt-out 98.9%

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

      6. div-inv 98.9%

         \[\leadsto \mathsf{fma}\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right), \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      7. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in l around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. associate-*r* 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-*r* 98.9%

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

      7. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if -2.5999999999999999e94 < l < -0.029999999999999999 or 1.3e13 < l < 2.02000000000000006e96

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 80.0%

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

   if -0.029999999999999999 < l < 1.3e13

   1. Initial program 68.9%

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

   2. Taylor expanded in l around 0 98.5%

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

      1. fma-def 98.5%

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

      2. +-commutative 98.5%

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

      3. associate-*r* 98.5%

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

      4. associate-*r* 98.5%

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

      5. distribute-rgt-out 98.5%

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

      6. div-inv 98.5%

         \[\leadsto \mathsf{fma}\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right), \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      7. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+94}:\\ \;\;\;\;\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -0.03:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 13000000000000:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(J \cdot 2\right), \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{+96}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.86:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.86)
     (+ U (* t_0 (* l (* J 2.0))))
     (+ U (+ (* 2.0 (* l J)) (* 0.3333333333333333 (* J (pow l 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.86) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + ((2.0 * (l * J)) + (0.3333333333333333 * (J * pow(l, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.86d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u + ((2.0d0 * (l * j)) + (0.3333333333333333d0 * (j * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.86) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + ((2.0 * (l * J)) + (0.3333333333333333 * (J * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.86:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + ((2.0 * (l * J)) + (0.3333333333333333 * (J * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.86)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(Float64(2.0 * Float64(l * J)) + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))));
	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 <= 0.86)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + ((2.0 * (l * J)) + (0.3333333333333333 * (J * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.86], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.859999999999999987

   1. Initial program 82.6%

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

   2. Taylor expanded in l around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l* 68.2%

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

   4. Simplified 68.2%

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

   if 0.859999999999999987 < (cos.f64 (/.f64 K 2))

   1. Initial program 84.7%

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

   2. Taylor expanded in l around 0 91.0%

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

   3. Taylor expanded in K around 0 89.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.86:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 6: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.86:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.86)
     (+ U (* t_0 (* l (* J 2.0))))
     (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.86) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.86d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.86) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.86:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.86)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.86)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.86], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.859999999999999987

   1. Initial program 82.6%

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

   2. Taylor expanded in l around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l* 68.2%

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

   4. Simplified 68.2%

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

   if 0.859999999999999987 < (cos.f64 (/.f64 K 2))

   1. Initial program 84.7%

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

   2. Taylor expanded in l around 0 91.0%

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

      1. fma-def 91.0%

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

      2. +-commutative 91.0%

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

      3. associate-*r* 91.0%

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

      4. associate-*r* 91.0%

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

      5. distribute-rgt-out 91.0%

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

      6. div-inv 91.0%

         \[\leadsto \mathsf{fma}\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right), \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      7. metadata-eval 91.0%

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in K around 0 89.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.86:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 7: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.03 \lor \neg \left(\ell \leq 13000000000000\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(J \cdot 2\right), \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.03) (not (<= l 13000000000000.0)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (fma (* l (* J 2.0)) (cos (* K 0.5)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.03) || !(l <= 13000000000000.0)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = fma((l * (J * 2.0)), cos((K * 0.5)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.03) || !(l <= 13000000000000.0))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = fma(Float64(l * Float64(J * 2.0)), cos(Float64(K * 0.5)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.03], N[Not[LessEqual[l, 13000000000000.0]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.029999999999999999 or 1.3e13 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 74.6%

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

   if -0.029999999999999999 < l < 1.3e13

   1. Initial program 68.9%

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

   2. Taylor expanded in l around 0 98.5%

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

      1. fma-def 98.5%

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

      2. +-commutative 98.5%

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

      3. associate-*r* 98.5%

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

      4. associate-*r* 98.5%

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

      5. distribute-rgt-out 98.5%

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

      6. div-inv 98.5%

         \[\leadsto \mathsf{fma}\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right), \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}, U\right) \]

      7. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.03 \lor \neg \left(\ell \leq 13000000000000\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \left(J \cdot 2\right), \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \]

Alternative 8: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.03 \lor \neg \left(\ell \leq 13000000000000\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.03) (not (<= l 13000000000000.0)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.03) || !(l <= 13000000000000.0)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.03d0)) .or. (.not. (l <= 13000000000000.0d0))) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.03) || !(l <= 13000000000000.0)) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.03) or not (l <= 13000000000000.0):
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.03) || !(l <= 13000000000000.0))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.03) || ~((l <= 13000000000000.0)))
		tmp = U + ((exp(l) - exp(-l)) * J);
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.03], N[Not[LessEqual[l, 13000000000000.0]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.029999999999999999 or 1.3e13 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 74.6%

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

   if -0.029999999999999999 < l < 1.3e13

   1. Initial program 68.9%

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

   2. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.03 \lor \neg \left(\ell \leq 13000000000000\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

Alternative 9: 64.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

2. Taylor expanded in l around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. associate-*l* 66.3%

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

   3. *-commutative 66.3%

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

4. Simplified 66.3%

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

5. Final simplification 66.3%

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

Alternative 10: 64.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

2. Taylor expanded in l around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. associate-*l* 66.3%

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

4. Simplified 66.3%

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

5. Final simplification 66.3%

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

Alternative 11: 47.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -700:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -700.0) (pow U -8.0) (if (<= l 1.3e-5) U (pow U -8.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -700.0) {
		tmp = pow(U, -8.0);
	} else if (l <= 1.3e-5) {
		tmp = U;
	} else {
		tmp = pow(U, -8.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-700.0d0)) then
        tmp = u ** (-8.0d0)
    else if (l <= 1.3d-5) then
        tmp = u
    else
        tmp = u ** (-8.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -700.0) {
		tmp = Math.pow(U, -8.0);
	} else if (l <= 1.3e-5) {
		tmp = U;
	} else {
		tmp = Math.pow(U, -8.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -700.0:
		tmp = math.pow(U, -8.0)
	elif l <= 1.3e-5:
		tmp = U
	else:
		tmp = math.pow(U, -8.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -700.0)
		tmp = U ^ -8.0;
	elseif (l <= 1.3e-5)
		tmp = U;
	else
		tmp = U ^ -8.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -700.0)
		tmp = U ^ -8.0;
	elseif (l <= 1.3e-5)
		tmp = U;
	else
		tmp = U ^ -8.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -700.0], N[Power[U, -8.0], $MachinePrecision], If[LessEqual[l, 1.3e-5], U, N[Power[U, -8.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -700 or 1.29999999999999992e-5 < l

   1. Initial program 99.9%

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

   3. Applied egg-rr 25.1%

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

   if -700 < l < 1.29999999999999992e-5

   1. Initial program 68.3%

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

   2. Taylor expanded in J around 0 67.5%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -700:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \]

Alternative 12: 42.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\frac{0.109375}{U \cdot U} - \frac{0.125}{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.2e+18)
   (* U U)
   (if (<= l 1.3e-5) U (- (/ 0.109375 (* U U)) (/ 0.125 U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e+18) {
		tmp = U * U;
	} else if (l <= 1.3e-5) {
		tmp = U;
	} else {
		tmp = (0.109375 / (U * U)) - (0.125 / U);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-4.2d+18)) then
        tmp = u * u
    else if (l <= 1.3d-5) then
        tmp = u
    else
        tmp = (0.109375d0 / (u * u)) - (0.125d0 / u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e+18) {
		tmp = U * U;
	} else if (l <= 1.3e-5) {
		tmp = U;
	} else {
		tmp = (0.109375 / (U * U)) - (0.125 / U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.2e+18:
		tmp = U * U
	elif l <= 1.3e-5:
		tmp = U
	else:
		tmp = (0.109375 / (U * U)) - (0.125 / U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.2e+18)
		tmp = Float64(U * U);
	elseif (l <= 1.3e-5)
		tmp = U;
	else
		tmp = Float64(Float64(0.109375 / Float64(U * U)) - Float64(0.125 / U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4.2e+18)
		tmp = U * U;
	elseif (l <= 1.3e-5)
		tmp = U;
	else
		tmp = (0.109375 / (U * U)) - (0.125 / U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.2e+18], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.3e-5], U, N[(N[(0.109375 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(0.125 / U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.2e18

   1. Initial program 100.0%

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

   3. Applied egg-rr 14.8%

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

   if -4.2e18 < l < 1.29999999999999992e-5

   1. Initial program 69.3%

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

   2. Taylor expanded in J around 0 65.6%

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

   if 1.29999999999999992e-5 < l

   1. Initial program 99.7%

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

   3. Applied egg-rr 1.4%

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

      1. associate-+r+ 1.4%

         \[\leadsto \frac{U}{\color{blue}{\left(U + -8 \cdot U\right) + U \cdot \left(-8 \cdot U\right)}} \]

      2. distribute-rgt1-in 1.4%

         \[\leadsto \frac{U}{\color{blue}{\left(-8 + 1\right) \cdot U} + U \cdot \left(-8 \cdot U\right)} \]

      3. *-commutative 1.4%

         \[\leadsto \frac{U}{\left(-8 + 1\right) \cdot U + \color{blue}{\left(-8 \cdot U\right) \cdot U}} \]

      4. distribute-rgt-out 1.4%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(\left(-8 + 1\right) + -8 \cdot U\right)}} \]

      5. associate-/r* 1.4%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{\left(-8 + 1\right) + -8 \cdot U}} \]

      6. *-inverses 1.4%

         \[\leadsto \frac{\color{blue}{1}}{\left(-8 + 1\right) + -8 \cdot U} \]

      7. +-commutative 1.4%

         \[\leadsto \frac{1}{\color{blue}{-8 \cdot U + \left(-8 + 1\right)}} \]

      8. *-commutative 1.4%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -8} + \left(-8 + 1\right)} \]

      9. metadata-eval 1.4%

         \[\leadsto \frac{1}{U \cdot -8 + \color{blue}{-7}} \]

   5. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -8 + -7}} \]

   6. Taylor expanded in U around inf 23.1%

      \[\leadsto \color{blue}{0.109375 \cdot \frac{1}{{U}^{2}} - 0.125 \cdot \frac{1}{U}} \]
   7. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto 0.109375 \cdot \frac{1}{\color{blue}{U \cdot U}} - 0.125 \cdot \frac{1}{U} \]

      2. associate-*r/ 23.1%

         \[\leadsto \color{blue}{\frac{0.109375 \cdot 1}{U \cdot U}} - 0.125 \cdot \frac{1}{U} \]

      3. metadata-eval 23.1%

         \[\leadsto \frac{\color{blue}{0.109375}}{U \cdot U} - 0.125 \cdot \frac{1}{U} \]

      4. associate-*r/ 23.1%

         \[\leadsto \frac{0.109375}{U \cdot U} - \color{blue}{\frac{0.125 \cdot 1}{U}} \]

      5. metadata-eval 23.1%

         \[\leadsto \frac{0.109375}{U \cdot U} - \frac{\color{blue}{0.125}}{U} \]

   8. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{0.109375}{U \cdot U} - \frac{0.125}{U}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\frac{0.109375}{U \cdot U} - \frac{0.125}{U}\\ \end{array} \]

Alternative 13: 42.2% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+51}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.4e+18) (* U U) (if (<= l 2.45e+51) U (* U (- 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.4e+18) {
		tmp = U * U;
	} else if (l <= 2.45e+51) {
		tmp = U;
	} else {
		tmp = U * (2.0 - U);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-5.4d+18)) then
        tmp = u * u
    else if (l <= 2.45d+51) then
        tmp = u
    else
        tmp = u * (2.0d0 - u)
    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+18) {
		tmp = U * U;
	} else if (l <= 2.45e+51) {
		tmp = U;
	} else {
		tmp = U * (2.0 - U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.4e+18:
		tmp = U * U
	elif l <= 2.45e+51:
		tmp = U
	else:
		tmp = U * (2.0 - U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.4e+18)
		tmp = Float64(U * U);
	elseif (l <= 2.45e+51)
		tmp = U;
	else
		tmp = Float64(U * Float64(2.0 - U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.4e+18)
		tmp = U * U;
	elseif (l <= 2.45e+51)
		tmp = U;
	else
		tmp = U * (2.0 - U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.4e+18], N[(U * U), $MachinePrecision], If[LessEqual[l, 2.45e+51], U, N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.4e18

   1. Initial program 100.0%

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

   3. Applied egg-rr 14.8%

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

   if -5.4e18 < l < 2.44999999999999992e51

   1. Initial program 71.3%

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

   2. Taylor expanded in J around 0 61.2%

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

   if 2.44999999999999992e51 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 19.5%

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

      1. fma-udef 19.5%

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

      2. +-commutative 19.5%

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

      3. associate-+r+ 19.5%

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

      4. count-2 19.5%

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

      5. distribute-rgt-out 19.5%

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

      6. unsub-neg 19.5%

         \[\leadsto U \cdot \color{blue}{\left(2 - U\right)} \]

   7. Simplified 19.5%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+51}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \end{array} \]

Alternative 14: 39.9% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+16}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -7.5e+16) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7.5e+16) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-7.5d+16)) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7.5e+16) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -7.5e+16:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -7.5e+16)
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -7.5e+16)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -7.5e+16], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -7.5e16

   1. Initial program 100.0%

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

   3. Applied egg-rr 14.8%

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

   if -7.5e16 < l

   1. Initial program 78.7%

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

   2. Taylor expanded in J around 0 46.3%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+16}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -0.14285714285714285 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -0.14285714285714285)

C

double code(double J, double l, double K, double U) {
	return -0.14285714285714285;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = -0.14285714285714285d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -0.14285714285714285;
}

Python

def code(J, l, K, U):
	return -0.14285714285714285

Julia

function code(J, l, K, U)
	return -0.14285714285714285
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -0.14285714285714285;
end

Wolfram

code[J_, l_, K_, U_] := -0.14285714285714285

Derivation

1. Initial program 83.7%

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

3. Applied egg-rr 2.2%

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

   1. associate-+r+ 2.2%

      \[\leadsto \frac{U}{\color{blue}{\left(U + -8 \cdot U\right) + U \cdot \left(-8 \cdot U\right)}} \]

   2. distribute-rgt1-in 2.2%

      \[\leadsto \frac{U}{\color{blue}{\left(-8 + 1\right) \cdot U} + U \cdot \left(-8 \cdot U\right)} \]

   3. *-commutative 2.2%

      \[\leadsto \frac{U}{\left(-8 + 1\right) \cdot U + \color{blue}{\left(-8 \cdot U\right) \cdot U}} \]

   4. distribute-rgt-out 2.2%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(\left(-8 + 1\right) + -8 \cdot U\right)}} \]

   5. associate-/r* 2.2%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{\left(-8 + 1\right) + -8 \cdot U}} \]

   6. *-inverses 2.2%

      \[\leadsto \frac{\color{blue}{1}}{\left(-8 + 1\right) + -8 \cdot U} \]

   7. +-commutative 2.2%

      \[\leadsto \frac{1}{\color{blue}{-8 \cdot U + \left(-8 + 1\right)}} \]

   8. *-commutative 2.2%

      \[\leadsto \frac{1}{\color{blue}{U \cdot -8} + \left(-8 + 1\right)} \]

   9. metadata-eval 2.2%

      \[\leadsto \frac{1}{U \cdot -8 + \color{blue}{-7}} \]

5. Simplified 2.2%

   \[\leadsto \color{blue}{\frac{1}{U \cdot -8 + -7}} \]

6. Taylor expanded in U around 0 2.5%

   \[\leadsto \color{blue}{-0.14285714285714285} \]

7. Final simplification 2.5%

   \[\leadsto -0.14285714285714285 \]

Alternative 16: 2.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Applied egg-rr 3.1%

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

   1. *-inverses 3.1%

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

5. Simplified 3.1%

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

6. Final simplification 3.1%

   \[\leadsto 1 \]

Alternative 17: 36.9% 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 83.7%

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

2. Taylor expanded in J around 0 35.7%

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

3. Final simplification 35.7%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023195 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))