Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.7%

Time: 9.4s

Alternatives: 12

Speedup: 0.5×

• 50.4% 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: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -10.0) (not (<= t_0 0.0)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+ U (* 2.0 (* l (* J (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 <= -10.0) || !(t_0 <= 0.0)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (2.0 * (l * (J * 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 <= (-10.0d0)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (cos((k / 2.0d0)) * (t_0 * j)) + u
    else
        tmp = u + (2.0d0 * (l * (j * 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 <= -10.0) || !(t_0 <= 0.0)) {
		tmp = (Math.cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (2.0 * (l * (J * 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 <= -10.0) or not (t_0 <= 0.0):
		tmp = (math.cos((K / 2.0)) * (t_0 * J)) + U
	else:
		tmp = U + (2.0 * (l * (J * 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 <= -10.0) || !(t_0 <= 0.0))
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * 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 <= -10.0) || ~((t_0 <= 0.0)))
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	else
		tmp = U + (2.0 * (l * (J * 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, -10.0], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -10 or 0.0 < (-.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 -10 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 69.9%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -10 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 2: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := t_0 \cdot J\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (* t_0 J)))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.0) (+ U (* 2.0 (* l (* J (cos (* K 0.5)))))) (+ U t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = t_0 * J;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else {
		tmp = U + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = t_0 * j
    if (t_0 <= (-10.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else
        tmp = u + t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double t_1 = t_0 * J;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = U + t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = t_0 * J
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	else:
		tmp = U + t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = Float64(t_0 * J)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = t_0 * J;
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	else
		tmp = U + t_1;
	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]}, Block[{t$95$1 = N[(t$95$0 * J), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -10

   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 84.6%

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

   3. Taylor expanded in J around inf 84.6%

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

   if -10 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 69.9%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 0.0 < (-.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 76.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -10:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -10.0) (not (<= t_0 2e-6)))
     (* t_0 J)
     (+ U (* 2.0 (* l (* J (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 <= -10.0) || !(t_0 <= 2e-6)) {
		tmp = t_0 * J;
	} else {
		tmp = U + (2.0 * (l * (J * 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 <= (-10.0d0)) .or. (.not. (t_0 <= 2d-6))) then
        tmp = t_0 * j
    else
        tmp = u + (2.0d0 * (l * (j * 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 <= -10.0) || !(t_0 <= 2e-6)) {
		tmp = t_0 * J;
	} else {
		tmp = U + (2.0 * (l * (J * 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 <= -10.0) or not (t_0 <= 2e-6):
		tmp = t_0 * J
	else:
		tmp = U + (2.0 * (l * (J * 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 <= -10.0) || !(t_0 <= 2e-6))
		tmp = Float64(t_0 * J);
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * 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 <= -10.0) || ~((t_0 <= 2e-6)))
		tmp = t_0 * J;
	else
		tmp = U + (2.0 * (l * (J * 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, -10.0], N[Not[LessEqual[t$95$0, 2e-6]], $MachinePrecision]], N[(t$95$0 * J), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -10 or 1.99999999999999991e-6 < (-.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 79.5%

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

   3. Taylor expanded in J around inf 79.5%

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

   if -10 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 70.6%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -10 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 4: 78.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.15:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.15)
   (+ U (* 2.0 (* l (* J (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)) <= 0.15) {
		tmp = U + (2.0 * (l * (J * 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)) <= 0.15d0) then
        tmp = u + (2.0d0 * (l * (j * 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)) <= 0.15) {
		tmp = U + (2.0 * (l * (J * 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)) <= 0.15:
		tmp = U + (2.0 * (l * (J * 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)) <= 0.15)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * 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)) <= 0.15)
		tmp = U + (2.0 * (l * (J * 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], 0.15], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $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)) < 0.149999999999999994

   1. Initial program 85.1%

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

   2. Taylor expanded in l around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*l* 65.9%

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

      3. *-commutative 65.9%

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

   4. Simplified 65.9%

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

   if 0.149999999999999994 < (cos.f64 (/.f64 K 2))

   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 K around 0 84.5%

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

   3. Taylor expanded in l around 0 88.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.15:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \lor \neg \left(\ell \leq 9 \cdot 10^{+41}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.6) (not (<= l 9e+41)))
   (* J (+ (* l 2.0) (* 0.3333333333333333 (* l (* l l)))))
   (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.6) || !(l <= 9e+41)) {
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	} else {
		tmp = U + (2.0 * (l * (J * 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) :: tmp
    if ((l <= (-2.6d0)) .or. (.not. (l <= 9d+41))) then
        tmp = j * ((l * 2.0d0) + (0.3333333333333333d0 * (l * (l * l))))
    else
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.6) || !(l <= 9e+41)) {
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.6) or not (l <= 9e+41):
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))))
	else:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.6) || !(l <= 9e+41))
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * Float64(l * Float64(l * l)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.6) || ~((l <= 9e+41)))
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	else
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.6], N[Not[LessEqual[l, 9e+41]], $MachinePrecision]], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.60000000000000009 or 9.0000000000000002e41 < 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 81.1%

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

   3. Taylor expanded in l around 0 72.5%

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

      1. associate-+r+ 72.5%

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

      2. associate-*r* 72.5%

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

      3. associate-*r* 72.5%

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

      4. distribute-rgt-out 72.5%

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

      5. fma-def 72.5%

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

      6. +-commutative 72.5%

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

      7. fma-def 72.5%

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

   5. Simplified 72.5%

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

   6. Taylor expanded in J around inf 72.6%

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

      1. unpow3 72.6%

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

   8. Applied egg-rr 72.6%

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

   if -2.60000000000000009 < l < 9.0000000000000002e41

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l* 92.7%

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

      3. *-commutative 92.7%

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

   4. Simplified 92.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \lor \neg \left(\ell \leq 9 \cdot 10^{+41}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 6: 71.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \lor \neg \left(\ell \leq 60000000000\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.6) (not (<= l 60000000000.0)))
   (* J (+ (* l 2.0) (* 0.3333333333333333 (* l (* l l)))))
   (fma (* l 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.6) || !(l <= 60000000000.0)) {
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	} else {
		tmp = fma((l * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.6) || !(l <= 60000000000.0))
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * Float64(l * Float64(l * l)))));
	else
		tmp = fma(Float64(l * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.6], N[Not[LessEqual[l, 60000000000.0]], $MachinePrecision]], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.60000000000000009 or 6e10 < 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 80.3%

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

   3. Taylor expanded in l around 0 68.9%

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

      1. associate-+r+ 68.9%

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

      2. associate-*r* 68.9%

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

      3. associate-*r* 68.9%

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

      4. distribute-rgt-out 68.9%

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

      5. fma-def 68.9%

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

      6. +-commutative 68.9%

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

      7. fma-def 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in J around inf 69.0%

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

      1. unpow3 69.0%

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

   8. Applied egg-rr 69.0%

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

   if -2.60000000000000009 < l < 6e10

   1. Initial program 71.6%

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

   2. Taylor expanded in K around 0 70.2%

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

   3. Taylor expanded in l around 0 84.6%

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

      1. associate-+r+ 84.6%

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

      2. associate-*r* 84.6%

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

      3. associate-*r* 84.6%

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

      4. distribute-rgt-out 84.6%

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

      5. fma-def 84.7%

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

      6. +-commutative 84.7%

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

      7. fma-def 84.7%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in l around 0 84.6%

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

      1. associate-*r* 84.6%

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

      2. *-commutative 84.6%

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

      3. fma-def 84.6%

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

   8. Simplified 84.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \lor \neg \left(\ell \leq 60000000000\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot 2, J, U\right)\\ \end{array} \]

Alternative 7: 71.0% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \lor \neg \left(\ell \leq 10^{+15}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.6) (not (<= l 1e+15)))
   (* J (+ (* l 2.0) (* 0.3333333333333333 (* l (* l l)))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.6) || !(l <= 1e+15)) {
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.6d0)) .or. (.not. (l <= 1d+15))) then
        tmp = j * ((l * 2.0d0) + (0.3333333333333333d0 * (l * (l * l))))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.6) || !(l <= 1e+15)) {
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.6) or not (l <= 1e+15):
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.6) || !(l <= 1e+15))
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * Float64(l * Float64(l * l)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.6) || ~((l <= 1e+15)))
		tmp = J * ((l * 2.0) + (0.3333333333333333 * (l * (l * l))));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.6], N[Not[LessEqual[l, 1e+15]], $MachinePrecision]], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.60000000000000009 or 1e15 < 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 80.3%

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

   3. Taylor expanded in l around 0 68.9%

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

      1. associate-+r+ 68.9%

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

      2. associate-*r* 68.9%

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

      3. associate-*r* 68.9%

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

      4. distribute-rgt-out 68.9%

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

      5. fma-def 68.9%

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

      6. +-commutative 68.9%

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

      7. fma-def 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in J around inf 69.0%

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

      1. unpow3 69.0%

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

   8. Applied egg-rr 69.0%

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

   if -2.60000000000000009 < l < 1e15

   1. Initial program 71.6%

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

   2. Taylor expanded in K around 0 70.2%

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

   3. Taylor expanded in l around 0 84.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \lor \neg \left(\ell \leq 10^{+15}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 42.9% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -9.2 \cdot 10^{+191} \lor \neg \left(J \leq 5.5 \cdot 10^{+120}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -9.2e+191) (not (<= J 5.5e+120))) (* J (* l 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -9.2e+191) || !(J <= 5.5e+120)) {
		tmp = J * (l * 2.0);
	} 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 ((j <= (-9.2d+191)) .or. (.not. (j <= 5.5d+120))) then
        tmp = j * (l * 2.0d0)
    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 ((J <= -9.2e+191) || !(J <= 5.5e+120)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -9.2e+191) or not (J <= 5.5e+120):
		tmp = J * (l * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -9.2e+191) || !(J <= 5.5e+120))
		tmp = Float64(J * Float64(l * 2.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -9.2e+191) || ~((J <= 5.5e+120)))
		tmp = J * (l * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -9.2e+191], N[Not[LessEqual[J, 5.5e+120]], $MachinePrecision]], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if J < -9.1999999999999997e191 or 5.50000000000000003e120 < J

   1. Initial program 66.2%

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

   2. Taylor expanded in K around 0 60.4%

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

   3. Taylor expanded in l around 0 82.5%

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

      1. associate-+r+ 82.5%

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

      2. associate-*r* 82.5%

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

      3. associate-*r* 82.5%

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

      4. distribute-rgt-out 82.5%

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

      5. fma-def 82.5%

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

      6. +-commutative 82.5%

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

      7. fma-def 82.5%

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

   5. Simplified 82.5%

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

   6. Taylor expanded in J around inf 68.9%

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

   7. Taylor expanded in l around 0 56.9%

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

      1. associate-*r* 56.9%

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

      2. *-commutative 56.9%

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

      3. *-commutative 56.9%

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

   9. Simplified 56.9%

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

   if -9.1999999999999997e191 < J < 5.50000000000000003e120

   1. Initial program 91.4%

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

   2. Taylor expanded in J around 0 46.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.2 \cdot 10^{+191} \lor \neg \left(J \leq 5.5 \cdot 10^{+120}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9: 41.1% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+40}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.6) (* U U) (if (<= l 6.4e+40) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.6) {
		tmp = U * U;
	} else if (l <= 6.4e+40) {
		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 <= (-2.6d0)) then
        tmp = u * u
    else if (l <= 6.4d+40) 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 <= -2.6) {
		tmp = U * U;
	} else if (l <= 6.4e+40) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.6:
		tmp = U * U
	elif l <= 6.4e+40:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.6)
		tmp = Float64(U * U);
	elseif (l <= 6.4e+40)
		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 <= -2.6)
		tmp = U * U;
	elseif (l <= 6.4e+40)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.6], N[(U * U), $MachinePrecision], If[LessEqual[l, 6.4e+40], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.60000000000000009 or 6.39999999999999961e40 < 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 12.4%

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

   if -2.60000000000000009 < l < 6.39999999999999961e40

   1. Initial program 72.4%

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

   2. Taylor expanded in J around 0 66.4%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+40}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 10: 53.5% 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 84.6%

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

2. Taylor expanded in K around 0 74.8%

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

3. Taylor expanded in l around 0 59.0%

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

4. Final simplification 59.0%

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

Alternative 11: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Applied egg-rr 2.6%

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

   1. associate-+r+ 2.6%

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

   2. distribute-rgt1-in 2.6%

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

   3. *-commutative 2.6%

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

   4. distribute-rgt-out 2.6%

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

   5. associate-/r* 2.6%

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

   6. *-inverses 2.6%

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

   7. +-commutative 2.6%

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

   8. *-commutative 2.6%

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

   9. metadata-eval 2.6%

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

5. Simplified 2.6%

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

6. Taylor expanded in U around 0 2.9%

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

7. Final simplification 2.9%

   \[\leadsto -0.14285714285714285 \]

Alternative 12: 35.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

2. Taylor expanded in J around 0 38.1%

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

3. Final simplification 38.1%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023203 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))