Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.8%

Time: 11.8s

Alternatives: 14

Speedup: 1.4×

• 49.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;J \cdot \left(t_0 \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-7)))
     (+ (* J (* t_0 (cos (* 0.5 K)))) U)
     (+
      U
      (*
       (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
       (cos (/ K 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-7)) {
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	} else {
		tmp = U + ((J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))) * cos((K / 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e-7)) {
		tmp = (J * (t_0 * Math.cos((0.5 * K)))) + U;
	} else {
		tmp = U + ((J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))) * Math.cos((K / 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-7):
		tmp = (J * (t_0 * math.cos((0.5 * K)))) + U
	else:
		tmp = U + ((J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))) * math.cos((K / 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-7))
		tmp = Float64(Float64(J * Float64(t_0 * cos(Float64(0.5 * K)))) + U);
	else
		tmp = Float64(U + Float64(Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))) * cos(Float64(K / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-7)))
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	else
		tmp = U + ((J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))) * cos((K / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-7]], $MachinePrecision]], N[(N[(J * N[(t$95$0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $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 4.99999999999999977e-7 < (-.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 J around 0 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.99999999999999977e-7

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 2: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;U + t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-7)))
     (+ U (* t_0 J))
     (+ U (* l (* 2.0 (* J (cos (* 0.5 K)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-7)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (l * (2.0 * (J * cos((0.5 * K)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e-7)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (l * (2.0 * (J * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-7):
		tmp = U + (t_0 * J)
	else:
		tmp = U + (l * (2.0 * (J * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-7))
		tmp = Float64(U + Float64(t_0 * J));
	else
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(0.5 * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-7)))
		tmp = U + (t_0 * J);
	else
		tmp = U + (l * (2.0 * (J * cos((0.5 * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-7]], $MachinePrecision]], N[(U + N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(0.5 * K), $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 4.99999999999999977e-7 < (-.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 72.1%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.99999999999999977e-7

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 3: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -350:\\ \;\;\;\;U + t_1 \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.21 \lor \neg \left(\ell \leq 4.5 \cdot 10^{+68}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + t_1 \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
           (cos (/ K 2.0)))))
        (t_1 (- (exp l) (exp (- l)))))
   (if (<= l -4.5e+76)
     t_0
     (if (<= l -350.0)
       (+ U (* t_1 (+ J (* J (* K (* K -0.125))))))
       (if (or (<= l 0.21) (not (<= l 4.5e+68))) t_0 (+ U (* t_1 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))) * cos((K / 2.0)));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (l <= -4.5e+76) {
		tmp = t_0;
	} else if (l <= -350.0) {
		tmp = U + (t_1 * (J + (J * (K * (K * -0.125)))));
	} else if ((l <= 0.21) || !(l <= 4.5e+68)) {
		tmp = t_0;
	} else {
		tmp = U + (t_1 * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + ((j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))) * cos((k / 2.0d0)))
    t_1 = exp(l) - exp(-l)
    if (l <= (-4.5d+76)) then
        tmp = t_0
    else if (l <= (-350.0d0)) then
        tmp = u + (t_1 * (j + (j * (k * (k * (-0.125d0))))))
    else if ((l <= 0.21d0) .or. (.not. (l <= 4.5d+68))) then
        tmp = t_0
    else
        tmp = u + (t_1 * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))) * Math.cos((K / 2.0)));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if (l <= -4.5e+76) {
		tmp = t_0;
	} else if (l <= -350.0) {
		tmp = U + (t_1 * (J + (J * (K * (K * -0.125)))));
	} else if ((l <= 0.21) || !(l <= 4.5e+68)) {
		tmp = t_0;
	} else {
		tmp = U + (t_1 * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))) * math.cos((K / 2.0)))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if l <= -4.5e+76:
		tmp = t_0
	elif l <= -350.0:
		tmp = U + (t_1 * (J + (J * (K * (K * -0.125)))))
	elif (l <= 0.21) or not (l <= 4.5e+68):
		tmp = t_0
	else:
		tmp = U + (t_1 * J)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))) * cos(Float64(K / 2.0))))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (l <= -4.5e+76)
		tmp = t_0;
	elseif (l <= -350.0)
		tmp = Float64(U + Float64(t_1 * Float64(J + Float64(J * Float64(K * Float64(K * -0.125))))));
	elseif ((l <= 0.21) || !(l <= 4.5e+68))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(t_1 * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))) * cos((K / 2.0)));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if (l <= -4.5e+76)
		tmp = t_0;
	elseif (l <= -350.0)
		tmp = U + (t_1 * (J + (J * (K * (K * -0.125)))));
	elseif ((l <= 0.21) || ~((l <= 4.5e+68)))
		tmp = t_0;
	else
		tmp = U + (t_1 * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+76], t$95$0, If[LessEqual[l, -350.0], N[(U + N[(t$95$1 * N[(J + N[(J * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 0.21], N[Not[LessEqual[l, 4.5e+68]], $MachinePrecision]], t$95$0, N[(U + N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.4999999999999997e76 or -350 < l < 0.209999999999999992 or 4.5000000000000003e68 < l

   1. Initial program 84.5%

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

   2. Taylor expanded in l around 0 98.3%

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

   if -4.4999999999999997e76 < l < -350

   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 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-*r* 85.7%

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

      6. *-commutative 85.7%

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

      7. associate-*l* 85.7%

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

      8. *-commutative 85.7%

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

      9. unpow2 85.7%

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

      10. associate-*l* 85.7%

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

   4. Simplified 85.7%

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

   if 0.209999999999999992 < l < 4.5000000000000003e68

   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 74.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+76}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq -350:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.21 \lor \neg \left(\ell \leq 4.5 \cdot 10^{+68}\right):\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]

Alternative 4: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.002)
   (+ U (* l (* 2.0 (* J (cos (* 0.5 K))))))
   (+ 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)) <= -0.002) {
		tmp = U + (l * (2.0 * (J * cos((0.5 * K)))));
	} 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)) <= (-0.002d0)) then
        tmp = u + (l * (2.0d0 * (j * cos((0.5d0 * k)))))
    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)) <= -0.002) {
		tmp = U + (l * (2.0 * (J * Math.cos((0.5 * K)))));
	} 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)) <= -0.002:
		tmp = U + (l * (2.0 * (J * math.cos((0.5 * K)))))
	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)) <= -0.002)
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(0.5 * K))))));
	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)) <= -0.002)
		tmp = U + (l * (2.0 * (J * cos((0.5 * K)))));
	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], -0.002], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(0.5 * K), $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)) < -2e-3

   1. Initial program 86.4%

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

   2. Taylor expanded in l around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. *-commutative 69.0%

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

      3. associate-*l* 69.0%

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

      4. *-commutative 69.0%

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

      5. associate-*l* 69.0%

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

   4. Simplified 69.0%

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

   if -2e-3 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.2%

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

   2. Taylor expanded in l around 0 90.2%

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

   3. Taylor expanded in K around 0 86.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.002:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 88.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + ((J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))) * cos((K / 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 * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))) * cos((k / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

2. Taylor expanded in l around 0 89.6%

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

3. Final simplification 89.6%

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

Alternative 6: 78.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+44} \lor \neg \left(\ell \leq 9 \cdot 10^{+50}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.8e+44) (not (<= l 9e+50)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* l (* 2.0 (* J (cos (* 0.5 K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.8e+44) || !(l <= 9e+50)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (l * (2.0 * (J * cos((0.5 * K)))));
	}
	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 <= (-2.8d+44)) .or. (.not. (l <= 9d+50))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (l * (2.0d0 * (j * cos((0.5d0 * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.8e+44) || !(l <= 9e+50)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (l * (2.0 * (J * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.8e+44) or not (l <= 9e+50):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (l * (2.0 * (J * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.8e+44) || !(l <= 9e+50))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(0.5 * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.8e+44) || ~((l <= 9e+50)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (l * (2.0 * (J * cos((0.5 * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.8e+44], N[Not[LessEqual[l, 9e+50]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.8000000000000001e44 or 9.00000000000000027e50 < 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 87.1%

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

   3. Taylor expanded in K around 0 65.9%

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

   4. Taylor expanded in l around inf 65.9%

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

   if -2.8000000000000001e44 < l < 9.00000000000000027e50

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 89.9%

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

      1. *-commutative 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*l* 89.9%

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

      4. *-commutative 89.9%

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

      5. associate-*l* 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+44} \lor \neg \left(\ell \leq 9 \cdot 10^{+50}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 7: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -340 \lor \neg \left(\ell \leq 2.5\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -340.0) (not (<= l 2.5)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -340.0) || !(l <= 2.5)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-340.0d0)) .or. (.not. (l <= 2.5d0))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (l * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -340.0) || !(l <= 2.5)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -340.0) or not (l <= 2.5):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -340.0) || !(l <= 2.5))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -340.0) || ~((l <= 2.5)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -340.0], N[Not[LessEqual[l, 2.5]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -340 or 2.5 < 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 78.0%

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

   3. Taylor expanded in K around 0 58.0%

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

   4. Taylor expanded in l around inf 58.0%

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

   if -340 < l < 2.5

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in K around 0 87.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -340 \lor \neg \left(\ell \leq 2.5\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 8: 60.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := K \cdot \left(\ell \cdot \left(K \cdot -0.25\right)\right)\\ t_1 := U + J \cdot \left(\ell \cdot \left(2 + K \cdot \left(K \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+153} \lor \neg \left(\ell \leq 4.6 \cdot 10^{+238}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \frac{t_0 \cdot t_0 - \left(\ell \cdot 2\right) \cdot \left(\ell \cdot 2\right)}{t_0 - \ell \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* K (* l (* K -0.25))))
        (t_1 (+ U (* J (* l (+ 2.0 (* K (* K -0.25))))))))
   (if (<= l -2.2e+34)
     t_1
     (if (<= l 200.0)
       (+ U (* l (* J 2.0)))
       (if (or (<= l 7e+153) (not (<= l 4.6e+238)))
         t_1
         (+
          U
          (*
           J
           (/ (- (* t_0 t_0) (* (* l 2.0) (* l 2.0))) (- t_0 (* l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = K * (l * (K * -0.25));
	double t_1 = U + (J * (l * (2.0 + (K * (K * -0.25)))));
	double tmp;
	if (l <= -2.2e+34) {
		tmp = t_1;
	} else if (l <= 200.0) {
		tmp = U + (l * (J * 2.0));
	} else if ((l <= 7e+153) || !(l <= 4.6e+238)) {
		tmp = t_1;
	} else {
		tmp = U + (J * (((t_0 * t_0) - ((l * 2.0) * (l * 2.0))) / (t_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 = k * (l * (k * (-0.25d0)))
    t_1 = u + (j * (l * (2.0d0 + (k * (k * (-0.25d0))))))
    if (l <= (-2.2d+34)) then
        tmp = t_1
    else if (l <= 200.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else if ((l <= 7d+153) .or. (.not. (l <= 4.6d+238))) then
        tmp = t_1
    else
        tmp = u + (j * (((t_0 * t_0) - ((l * 2.0d0) * (l * 2.0d0))) / (t_0 - (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 = K * (l * (K * -0.25));
	double t_1 = U + (J * (l * (2.0 + (K * (K * -0.25)))));
	double tmp;
	if (l <= -2.2e+34) {
		tmp = t_1;
	} else if (l <= 200.0) {
		tmp = U + (l * (J * 2.0));
	} else if ((l <= 7e+153) || !(l <= 4.6e+238)) {
		tmp = t_1;
	} else {
		tmp = U + (J * (((t_0 * t_0) - ((l * 2.0) * (l * 2.0))) / (t_0 - (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = K * (l * (K * -0.25))
	t_1 = U + (J * (l * (2.0 + (K * (K * -0.25)))))
	tmp = 0
	if l <= -2.2e+34:
		tmp = t_1
	elif l <= 200.0:
		tmp = U + (l * (J * 2.0))
	elif (l <= 7e+153) or not (l <= 4.6e+238):
		tmp = t_1
	else:
		tmp = U + (J * (((t_0 * t_0) - ((l * 2.0) * (l * 2.0))) / (t_0 - (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(K * Float64(l * Float64(K * -0.25)))
	t_1 = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(K * Float64(K * -0.25))))))
	tmp = 0.0
	if (l <= -2.2e+34)
		tmp = t_1;
	elseif (l <= 200.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif ((l <= 7e+153) || !(l <= 4.6e+238))
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(t_0 * t_0) - Float64(Float64(l * 2.0) * Float64(l * 2.0))) / Float64(t_0 - Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = K * (l * (K * -0.25));
	t_1 = U + (J * (l * (2.0 + (K * (K * -0.25)))));
	tmp = 0.0;
	if (l <= -2.2e+34)
		tmp = t_1;
	elseif (l <= 200.0)
		tmp = U + (l * (J * 2.0));
	elseif ((l <= 7e+153) || ~((l <= 4.6e+238)))
		tmp = t_1;
	else
		tmp = U + (J * (((t_0 * t_0) - ((l * 2.0) * (l * 2.0))) / (t_0 - (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(K * N[(l * N[(K * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(l * N[(2.0 + N[(K * N[(K * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.2e+34], t$95$1, If[LessEqual[l, 200.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 7e+153], N[Not[LessEqual[l, 4.6e+238]], $MachinePrecision]], t$95$1, N[(U + N[(J * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(l * 2.0), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2000000000000002e34 or 200 < l < 6.9999999999999998e153 or 4.60000000000000005e238 < 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 25.7%

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

      1. *-commutative 25.7%

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

      2. *-commutative 25.7%

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

      3. associate-*l* 25.7%

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

      4. *-commutative 25.7%

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

      5. associate-*l* 25.7%

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

   4. Simplified 25.7%

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

   5. Taylor expanded in K around 0 26.8%

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

   6. Taylor expanded in J around 0 39.9%

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

      1. associate-*r* 39.9%

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

      2. distribute-rgt-out 39.9%

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

      3. unpow2 39.9%

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

      4. associate-*r* 39.9%

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

   8. Simplified 39.9%

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

   if -2.2000000000000002e34 < l < 200

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*l* 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*l* 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in K around 0 85.1%

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

   if 6.9999999999999998e153 < l < 4.60000000000000005e238

   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 27.4%

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

      1. *-commutative 27.4%

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

      2. *-commutative 27.4%

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

      3. associate-*l* 27.4%

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

      4. *-commutative 27.4%

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

      5. associate-*l* 27.4%

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

   4. Simplified 27.4%

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

   5. Taylor expanded in K around 0 21.1%

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

   6. Taylor expanded in J around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. distribute-rgt-out 32.8%

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

      3. unpow2 32.8%

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

      4. associate-*r* 32.8%

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

   8. Simplified 32.8%

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

      1. distribute-lft-in 32.8%

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

      2. flip-+ 58.8%

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

      3. *-commutative 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-*l* 58.8%

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

      6. *-commutative 58.8%

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

      7. *-commutative 58.8%

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

      8. associate-*l* 58.8%

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

      9. *-commutative 58.8%

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

      10. *-commutative 58.8%

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

      11. associate-*l* 58.8%

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

   10. Applied egg-rr 58.8%

       \[\leadsto J \cdot \color{blue}{\frac{\left(K \cdot \left(\left(-0.25 \cdot K\right) \cdot \ell\right)\right) \cdot \left(K \cdot \left(\left(-0.25 \cdot K\right) \cdot \ell\right)\right) - \left(\ell \cdot 2\right) \cdot \left(\ell \cdot 2\right)}{K \cdot \left(\left(-0.25 \cdot K\right) \cdot \ell\right) - \ell \cdot 2}} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+34}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + K \cdot \left(K \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+153} \lor \neg \left(\ell \leq 4.6 \cdot 10^{+238}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + K \cdot \left(K \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \frac{\left(K \cdot \left(\ell \cdot \left(K \cdot -0.25\right)\right)\right) \cdot \left(K \cdot \left(\ell \cdot \left(K \cdot -0.25\right)\right)\right) - \left(\ell \cdot 2\right) \cdot \left(\ell \cdot 2\right)}{K \cdot \left(\ell \cdot \left(K \cdot -0.25\right)\right) - \ell \cdot 2}\\ \end{array} \]

Alternative 9: 60.1% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+34} \lor \neg \left(\ell \leq 490\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + K \cdot \left(K \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.2e+34) (not (<= l 490.0)))
   (+ U (* J (* l (+ 2.0 (* K (* K -0.25))))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.2e+34) || !(l <= 490.0)) {
		tmp = U + (J * (l * (2.0 + (K * (K * -0.25)))));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.2d+34)) .or. (.not. (l <= 490.0d0))) then
        tmp = u + (j * (l * (2.0d0 + (k * (k * (-0.25d0))))))
    else
        tmp = u + (l * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.2e+34) || !(l <= 490.0)) {
		tmp = U + (J * (l * (2.0 + (K * (K * -0.25)))));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.2e+34) or not (l <= 490.0):
		tmp = U + (J * (l * (2.0 + (K * (K * -0.25)))))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.2e+34) || !(l <= 490.0))
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(K * Float64(K * -0.25))))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.2e+34) || ~((l <= 490.0)))
		tmp = U + (J * (l * (2.0 + (K * (K * -0.25)))));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.2e+34], N[Not[LessEqual[l, 490.0]], $MachinePrecision]], N[(U + N[(J * N[(l * N[(2.0 + N[(K * N[(K * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.2000000000000002e34 or 490 < 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 25.9%

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

      1. *-commutative 25.9%

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

      2. *-commutative 25.9%

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

      3. associate-*l* 25.9%

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

      4. *-commutative 25.9%

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

      5. associate-*l* 25.9%

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

   4. Simplified 25.9%

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

   5. Taylor expanded in K around 0 25.9%

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

   6. Taylor expanded in J around 0 38.9%

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

      1. associate-*r* 38.9%

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

      2. distribute-rgt-out 38.9%

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

      3. unpow2 38.9%

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

      4. associate-*r* 38.9%

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

   8. Simplified 38.9%

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

   if -2.2000000000000002e34 < l < 490

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*l* 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*l* 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in K around 0 85.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+34} \lor \neg \left(\ell \leq 490\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + K \cdot \left(K \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 10: 52.2% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + -0.25 \cdot \left(J \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= U 3.6e+182)
   (+ U (* l (* J 2.0)))
   (+ U (* -0.25 (* J (* l (* K K)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (U <= 3.6e+182) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (-0.25 * (J * (l * (K * K))));
	}
	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 (u <= 3.6d+182) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u + ((-0.25d0) * (j * (l * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (U <= 3.6e+182) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (-0.25 * (J * (l * (K * K))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if U <= 3.6e+182:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U + (-0.25 * (J * (l * (K * K))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (U <= 3.6e+182)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U + Float64(-0.25 * Float64(J * Float64(l * Float64(K * K)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (U <= 3.6e+182)
		tmp = U + (l * (J * 2.0));
	else
		tmp = U + (-0.25 * (J * (l * (K * K))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[U, 3.6e+182], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(-0.25 * N[(J * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 3.6e182

   1. Initial program 84.6%

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

   2. Taylor expanded in l around 0 65.0%

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

      1. *-commutative 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-*l* 65.0%

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

      4. *-commutative 65.0%

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

      5. associate-*l* 65.0%

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

   4. Simplified 65.0%

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

   5. Taylor expanded in K around 0 54.7%

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

   if 3.6e182 < U

   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 64.1%

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

      1. *-commutative 64.1%

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

      2. *-commutative 64.1%

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

      3. associate-*l* 64.1%

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

      4. *-commutative 64.1%

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

      5. associate-*l* 64.1%

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

   4. Simplified 64.1%

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

   5. Taylor expanded in K around 0 59.8%

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

   6. Taylor expanded in K around inf 71.1%

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

      1. *-commutative 71.1%

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

      2. unpow2 71.1%

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

   8. Simplified 71.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + -0.25 \cdot \left(J \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 11: 53.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

2. Taylor expanded in l around 0 64.9%

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

   1. *-commutative 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 64.9%

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

   4. *-commutative 64.9%

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

   5. associate-*l* 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in K around 0 54.1%

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

6. Final simplification 54.1%

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

Alternative 12: 40.1% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 280000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 280000.0) U (* U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 280000.0) {
		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 <= 280000.0d0) 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 <= 280000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 280000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 280000.0)
		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 <= 280000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 280000.0], U, N[(U * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.8e5

   1. Initial program 82.4%

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

      1. associate-*l* 82.4%

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

      2. fma-def 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in J around 0 49.8%

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

   if 2.8e5 < 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. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 15.7%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 280000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 13: 2.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate-*l* 86.2%

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

   2. fma-def 86.2%

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

3. Simplified 86.2%

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

5. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{\frac{-8 - U}{-8 - U}} \]
6. Step-by-step derivation

   1. *-inverses 2.8%

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

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto 1 \]

Alternative 14: 37.0% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate-*l* 86.2%

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

   2. fma-def 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in J around 0 39.5%

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

5. Final simplification 39.5%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023279 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))