Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.8%

Time: 13.2s

Alternatives: 16

Speedup: 1.5×

• 49.6% 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: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -1 \lor \neg \left(t_1 \leq 10^{-11}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -1.0) (not (<= t_1 1e-11)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* 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 t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -1.0) || !(t_1 <= 1e-11)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (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) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-1.0d0)) .or. (.not. (t_1 <= 1d-11))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (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 t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -1.0) || !(t_1 <= 1e-11)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (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))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -1.0) or not (t_1 <= 1e-11):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (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))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -1.0) || !(t_1 <= 1e-11))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * 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));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -1.0) || ~((t_1 <= 1e-11)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (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]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1.0], N[Not[LessEqual[t$95$1, 1e-11]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1 or 9.99999999999999939e-12 < (-.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 -1 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 9.99999999999999939e-12

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 10^{-11}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 10^{-11}\right):\\ \;\;\;\;t_0 \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -1.0) (not (<= t_0 1e-11)))
     (+ (* t_0 J) U)
     (+ U (* (* l J) (* 2.0 (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 1e-11)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 1d-11))) then
        tmp = (t_0 * j) + u
    else
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 1e-11)) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 1e-11):
		tmp = (t_0 * J) + U
	else:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 1e-11))
		tmp = Float64(Float64(t_0 * J) + U);
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 1e-11)))
		tmp = (t_0 * J) + U;
	else
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1.0], N[Not[LessEqual[t$95$0, 1e-11]], $MachinePrecision]], N[(N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1 or 9.99999999999999939e-12 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 70.0%

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

   if -1 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 9.99999999999999939e-12

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 10^{-11}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq -0.0265:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.000118:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot t_1, U\right)\\ \mathbf{elif}\;\ell \leq 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U)) (t_1 (cos (* K 0.5))))
   (if (<= l -4.2e+75)
     (* (* J 0.3333333333333333) (* (pow l 3.0) t_1))
     (if (<= l -0.0265)
       t_0
       (if (<= l 0.000118)
         (fma (* l J) (* 2.0 t_1) U)
         (if (<= l 1e+67)
           t_0
           (+
            U
            (* (cos (/ K 2.0)) (* 0.3333333333333333 (* J (pow l 3.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -4.2e+75) {
		tmp = (J * 0.3333333333333333) * (pow(l, 3.0) * t_1);
	} else if (l <= -0.0265) {
		tmp = t_0;
	} else if (l <= 0.000118) {
		tmp = fma((l * J), (2.0 * t_1), U);
	} else if (l <= 1e+67) {
		tmp = t_0;
	} else {
		tmp = U + (cos((K / 2.0)) * (0.3333333333333333 * (J * pow(l, 3.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -4.2e+75)
		tmp = Float64(Float64(J * 0.3333333333333333) * Float64((l ^ 3.0) * t_1));
	elseif (l <= -0.0265)
		tmp = t_0;
	elseif (l <= 0.000118)
		tmp = fma(Float64(l * J), Float64(2.0 * t_1), U);
	elseif (l <= 1e+67)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.2e+75], N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -0.0265], t$95$0, If[LessEqual[l, 0.000118], N[(N[(l * J), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1e+67], 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]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.19999999999999997e75

   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%

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

   4. Taylor expanded in l around 0 96.7%

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

   5. Taylor expanded in l around inf 96.8%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

   7. Simplified 96.8%

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

   if -4.19999999999999997e75 < l < -0.0264999999999999993 or 1.18e-4 < l < 9.99999999999999983e66

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 85.2%

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

   if -0.0264999999999999993 < l < 1.18e-4

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 9.99999999999999983e66 < 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 92.9%

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

   3. Taylor expanded in l around inf 92.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0265:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.000118:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 10^{+67}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 4: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot t_1\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -0.0055:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot t_1, U\right)\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U))
        (t_1 (cos (* K 0.5)))
        (t_2 (* (* J 0.3333333333333333) (* (pow l 3.0) t_1))))
   (if (<= l -4.2e+75)
     t_2
     (if (<= l -0.0055)
       t_0
       (if (<= l 0.00062)
         (fma (* l J) (* 2.0 t_1) U)
         (if (<= l 8.8e+66) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double t_2 = (J * 0.3333333333333333) * (pow(l, 3.0) * t_1);
	double tmp;
	if (l <= -4.2e+75) {
		tmp = t_2;
	} else if (l <= -0.0055) {
		tmp = t_0;
	} else if (l <= 0.00062) {
		tmp = fma((l * J), (2.0 * t_1), U);
	} else if (l <= 8.8e+66) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	t_2 = Float64(Float64(J * 0.3333333333333333) * Float64((l ^ 3.0) * t_1))
	tmp = 0.0
	if (l <= -4.2e+75)
		tmp = t_2;
	elseif (l <= -0.0055)
		tmp = t_0;
	elseif (l <= 0.00062)
		tmp = fma(Float64(l * J), Float64(2.0 * t_1), U);
	elseif (l <= 8.8e+66)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2e+75], t$95$2, If[LessEqual[l, -0.0055], t$95$0, If[LessEqual[l, 0.00062], N[(N[(l * J), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 8.8e+66], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.19999999999999997e75 or 8.7999999999999994e66 < 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%

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

   4. Taylor expanded in l around 0 94.2%

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

   5. Taylor expanded in l around inf 94.2%

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

      1. associate-*r* 94.2%

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

      2. *-commutative 94.2%

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

   7. Simplified 94.2%

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

   if -4.19999999999999997e75 < l < -0.0054999999999999997 or 6.2e-4 < l < 8.7999999999999994e66

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 85.2%

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

   if -0.0054999999999999997 < l < 6.2e-4

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0055:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+66}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 76.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ U (* (* l 2.0) (+ J (* -0.125 (* J (pow K 2.0))))))
   (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * 2.0) * (J + (-0.125 * (J * pow(K, 2.0)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + ((l * 2.0d0) * (j + ((-0.125d0) * (j * (k ** 2.0d0)))))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * 2.0) * (J + (-0.125 * (J * Math.pow(K, 2.0)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + ((l * 2.0) * (J + (-0.125 * (J * math.pow(K, 2.0)))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J + Float64(-0.125 * Float64(J * (K ^ 2.0))))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + ((l * 2.0) * (J + (-0.125 * (J * (K ^ 2.0)))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J + N[(-0.125 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.0200000000000000004

   1. Initial program 91.8%

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

   2. Taylor expanded in l around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. associate-*l* 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-*r* 59.0%

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

   4. Simplified 59.0%

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

   5. Taylor expanded in K around 0 60.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0 90.7%

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

   3. Taylor expanded in K around 0 88.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 6: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ U (* (* l J) (* 2.0 (cos (* K 0.5)))))
   (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * J) * (2.0 * Math.cos((K * 0.5))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + ((l * J) * (2.0 * math.cos((K * 0.5))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + ((l * J) * (2.0 * cos((K * 0.5))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $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)) < -0.0200000000000000004

   1. Initial program 91.8%

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

   2. Taylor expanded in l around 0 59.0%

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

      1. associate-*r* 59.0%

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

      2. associate-*r* 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-*l* 59.0%

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

      5. *-commutative 59.0%

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

   4. Simplified 59.0%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0 90.7%

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

   3. Taylor expanded in K around 0 88.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 7: 88.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (* (cos (/ K 2.0)) (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
end

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in l around 0 90.7%

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

3. Final simplification 90.7%

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

Alternative 8: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0245 \lor \neg \left(\ell \leq 0.0023\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0245) (not (<= l 0.0023)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (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 <= -0.0245) || !(l <= 0.0023)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} 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 <= -0.0245) || !(l <= 0.0023))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = fma(Float64(l * J), Float64(2.0 * cos(Float64(K * 0.5))), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0245], N[Not[LessEqual[l, 0.0023]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(l * J), $MachinePrecision] * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.024500000000000001 or 0.0023 < 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 70.0%

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

   if -0.024500000000000001 < l < 0.0023

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0245 \lor \neg \left(\ell \leq 0.0023\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot J, 2 \cdot \cos \left(K \cdot 0.5\right), U\right)\\ \end{array} \]

Alternative 9: 64.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + ((l * 2.0) * (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 + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in l around 0 62.3%

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

   1. *-commutative 62.3%

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

   2. associate-*l* 62.3%

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

   3. *-commutative 62.3%

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

   4. associate-*r* 62.3%

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

4. Simplified 62.3%

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

5. Final simplification 62.3%

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

Alternative 10: 64.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + ((l * J) * (2.0 * 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 + ((l * j) * (2.0d0 * cos((k * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in l around 0 62.3%

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

   1. associate-*r* 62.3%

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

   2. associate-*r* 62.3%

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

   3. *-commutative 62.3%

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

   4. associate-*l* 62.3%

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

   5. *-commutative 62.3%

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

4. Simplified 62.3%

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

5. Final simplification 62.3%

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

Alternative 11: 53.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+50}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 760:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{U} \cdot \frac{-4}{U} - U \cdot U}{U + \frac{-4}{U}}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.02e+50)
   (pow U -3.0)
   (if (<= l 760.0)
     (+ U (* J (* l 2.0)))
     (/ (- (* (/ -4.0 U) (/ -4.0 U)) (* U U)) (+ U (/ -4.0 U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.02e+50) {
		tmp = pow(U, -3.0);
	} else if (l <= 760.0) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = (((-4.0 / U) * (-4.0 / U)) - (U * U)) / (U + (-4.0 / 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 <= (-1.02d+50)) then
        tmp = u ** (-3.0d0)
    else if (l <= 760.0d0) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = ((((-4.0d0) / u) * ((-4.0d0) / u)) - (u * u)) / (u + ((-4.0d0) / u))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.02e+50) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 760.0) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = (((-4.0 / U) * (-4.0 / U)) - (U * U)) / (U + (-4.0 / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.02e+50:
		tmp = math.pow(U, -3.0)
	elif l <= 760.0:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = (((-4.0 / U) * (-4.0 / U)) - (U * U)) / (U + (-4.0 / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.02e+50)
		tmp = U ^ -3.0;
	elseif (l <= 760.0)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 / U) * Float64(-4.0 / U)) - Float64(U * U)) / Float64(U + Float64(-4.0 / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.02e+50)
		tmp = U ^ -3.0;
	elseif (l <= 760.0)
		tmp = U + (J * (l * 2.0));
	else
		tmp = (((-4.0 / U) * (-4.0 / U)) - (U * U)) / (U + (-4.0 / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.02e+50], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 760.0], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 / U), $MachinePrecision] * N[(-4.0 / U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-4.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.01999999999999991e50

   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 33.4%

      \[\leadsto \color{blue}{{U}^{-3}} \]

   if -1.01999999999999991e50 < l < 760

   1. Initial program 75.9%

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

   2. Taylor expanded in l around 0 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*l* 94.0%

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

      3. *-commutative 94.0%

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

      4. associate-*r* 94.0%

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

   4. Simplified 94.0%

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

   5. Taylor expanded in K around 0 84.1%

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

   if 760 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 3.2%

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

      1. sub-neg 3.2%

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

      2. flip-+ 28.8%

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

      3. frac-times 28.8%

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

      4. metadata-eval 28.8%

         \[\leadsto \frac{\frac{\color{blue}{16}}{U \cdot U} - \left(-U\right) \cdot \left(-U\right)}{\frac{-4}{U} - \left(-U\right)} \]

      5. pow2 28.8%

         \[\leadsto \frac{\frac{16}{\color{blue}{{U}^{2}}} - \left(-U\right) \cdot \left(-U\right)}{\frac{-4}{U} - \left(-U\right)} \]

   5. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{16}{{U}^{2}} - \left(-U\right) \cdot \left(-U\right)}{\frac{-4}{U} - \left(-U\right)}} \]
   6. Step-by-step derivation

      1. metadata-eval 28.8%

         \[\leadsto \frac{\frac{\color{blue}{-4 \cdot -4}}{{U}^{2}} - \left(-U\right) \cdot \left(-U\right)}{\frac{-4}{U} - \left(-U\right)} \]

      2. unpow2 28.8%

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

      3. frac-times 28.8%

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

   7. Applied egg-rr 28.8%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+50}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 760:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{U} \cdot \frac{-4}{U} - U \cdot U}{U + \frac{-4}{U}}\\ \end{array} \]

Alternative 12: 42.7% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+53} \lor \neg \left(\ell \leq 8500000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.1e+53) (not (<= l 8500000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e+53) || !(l <= 8500000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.1d+53)) .or. (.not. (l <= 8500000000.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e+53) || !(l <= 8500000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.1e+53) or not (l <= 8500000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.1e+53) || !(l <= 8500000000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.1e+53) || ~((l <= 8500000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.1e+53], N[Not[LessEqual[l, 8500000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2.1000000000000002e53 or 8.5e9 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 14.3%

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

   if -2.1000000000000002e53 < l < 8.5e9

   1. Initial program 76.6%

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

   2. Taylor expanded in J around 0 64.6%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+53} \lor \neg \left(\ell \leq 8500000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 42.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{elif}\;\ell \leq 23000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -7.5e-26) (* U (- 2.0 U)) (if (<= l 23000000000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7.5e-26) {
		tmp = U * (2.0 - U);
	} else if (l <= 23000000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-7.5d-26)) then
        tmp = u * (2.0d0 - u)
    else if (l <= 23000000000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7.5e-26) {
		tmp = U * (2.0 - U);
	} else if (l <= 23000000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -7.5e-26:
		tmp = U * (2.0 - U)
	elif l <= 23000000000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -7.5e-26)
		tmp = Float64(U * Float64(2.0 - U));
	elseif (l <= 23000000000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -7.5e-26)
		tmp = U * (2.0 - U);
	elseif (l <= 23000000000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -7.5e-26], N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 23000000000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.4999999999999994e-26

   1. Initial program 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-*l* 95.0%

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

      3. fma-def 95.0%

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

   3. Simplified 95.0%

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

   5. Applied egg-rr 13.1%

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

      1. fma-udef 13.1%

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

      2. +-commutative 13.1%

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

      3. associate-+r+ 13.1%

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

      4. count-2 13.1%

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

      5. distribute-rgt-out 13.1%

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

      6. unsub-neg 13.1%

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

   7. Simplified 13.1%

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

   if -7.4999999999999994e-26 < l < 2.3e10

   1. Initial program 76.6%

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

   2. Taylor expanded in J around 0 74.0%

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

   if 2.3e10 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 15.8%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{elif}\;\ell \leq 23000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 14: 54.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in l around 0 62.3%

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

   1. *-commutative 62.3%

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

   2. associate-*l* 62.3%

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

   3. *-commutative 62.3%

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

   4. associate-*r* 62.3%

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

4. Simplified 62.3%

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

5. Taylor expanded in K around 0 53.2%

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

6. Final simplification 53.2%

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

Alternative 15: 2.7% 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 88.1%

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

3. Applied egg-rr 2.5%

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

   1. *-inverses 2.5%

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

5. Simplified 2.5%

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

6. Final simplification 2.5%

   \[\leadsto 1 \]

Alternative 16: 37.3% 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 88.1%

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

2. Taylor expanded in J around 0 33.8%

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

3. Final simplification 33.8%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023309 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))