Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.6%

Time: 11.5s

Alternatives: 19

Speedup: 1.0×

• 50.2% 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: 86.5% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e-13)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e-13)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e-13)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e-13):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e-13))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e-13)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e-13]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * 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 4.9999999999999999e-13 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.9999999999999999e-13

   1. Initial program 67.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := t_0 \cdot J\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (* t_0 J)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 5e-13) (+ U (* (cos (/ K 2.0)) (* J (* l 2.0)))) (+ t_1 U)))))

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 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = t_1 + U;
	}
	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 t_1 = t_0 * J;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = t_1 + U;
	}
	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 <= -math.inf:
		tmp = t_1
	elif t_0 <= 5e-13:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = t_1 + U
	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 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(t_1 + U);
	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 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = t_1 + U;
	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, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 5e-13], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   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 75.4%

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

   3. Taylor expanded in J around inf 75.4%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.9999999999999999e-13

   1. Initial program 67.7%

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

   2. Taylor expanded in l around 0 99.9%

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

   if 4.9999999999999999e-13 < (-.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 80.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]

Alternative 3: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;U + t_0 \cdot \left(J + K \cdot \left(-0.125 \cdot \left(J \cdot K\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (<= t_0 (- INFINITY))
     (+ U (* t_0 (+ J (* K (* -0.125 (* J K))))))
     (if (<= t_0 5e-13)
       (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
       (+ (* t_0 J) U)))))

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)) {
		tmp = U + (t_0 * (J + (K * (-0.125 * (J * K)))));
	} else if (t_0 <= 5e-13) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = (t_0 * J) + U;
	}
	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) {
		tmp = U + (t_0 * (J + (K * (-0.125 * (J * K)))));
	} else if (t_0 <= 5e-13) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = (t_0 * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = U + (t_0 * (J + (K * (-0.125 * (J * K)))))
	elif t_0 <= 5e-13:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = (t_0 * J) + U
	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))
		tmp = Float64(U + Float64(t_0 * Float64(J + Float64(K * Float64(-0.125 * Float64(J * K))))));
	elseif (t_0 <= 5e-13)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(Float64(t_0 * J) + U);
	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)
		tmp = U + (t_0 * (J + (K * (-0.125 * (J * K)))));
	elseif (t_0 <= 5e-13)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = (t_0 * J) + U;
	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[LessEqual[t$95$0, (-Infinity)], N[(U + N[(t$95$0 * N[(J + N[(K * N[(-0.125 * N[(J * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-13], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   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 80.3%

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

      5. unpow2 80.3%

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

      6. associate-*r* 80.3%

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

      7. associate-*r* 80.3%

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

   4. Simplified 80.3%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.9999999999999999e-13

   1. Initial program 67.7%

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

   2. Taylor expanded in l around 0 99.9%

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

   if 4.9999999999999999e-13 < (-.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 80.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + K \cdot \left(-0.125 \cdot \left(J \cdot K\right)\right)\right)\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]

Alternative 4: 86.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-10)))
     (* t_0 J)
     (+ U (* (cos (/ K 2.0)) (* J (* l 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-10)) {
		tmp = t_0 * J;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * 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-10)) {
		tmp = t_0 * J;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 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-10):
		tmp = t_0 * J
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 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-10))
		tmp = Float64(t_0 * J);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 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-10)))
		tmp = t_0 * J;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 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-10]], $MachinePrecision]], N[(t$95$0 * J), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 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 5.00000000000000031e-10 < (-.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 78.2%

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

   3. Taylor expanded in J around inf 78.2%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.00000000000000031e-10

   1. Initial program 67.9%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 88.6%

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

Alternative 5: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;\ell \leq -110:\\ \;\;\;\;U + t_0 \cdot \left(J + K \cdot \left(-0.125 \cdot \left(J \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;t_0 \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \sqrt[3]{{\left(J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (<= l -110.0)
     (+ U (* t_0 (+ J (* K (* -0.125 (* J K))))))
     (if (<= l 2.1e-10)
       (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
       (if (<= l 6.5e+153)
         (+ (* t_0 J) U)
         (+ U (cbrt (pow (* J (* 2.0 (* l (cos (* K 0.5))))) 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if (l <= -110.0) {
		tmp = U + (t_0 * (J + (K * (-0.125 * (J * K)))));
	} else if (l <= 2.1e-10) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 6.5e+153) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + cbrt(pow((J * (2.0 * (l * cos((K * 0.5))))), 3.0));
	}
	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 (l <= -110.0) {
		tmp = U + (t_0 * (J + (K * (-0.125 * (J * K)))));
	} else if (l <= 2.1e-10) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 6.5e+153) {
		tmp = (t_0 * J) + U;
	} else {
		tmp = U + Math.cbrt(Math.pow((J * (2.0 * (l * Math.cos((K * 0.5))))), 3.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (l <= -110.0)
		tmp = Float64(U + Float64(t_0 * Float64(J + Float64(K * Float64(-0.125 * Float64(J * K))))));
	elseif (l <= 2.1e-10)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 6.5e+153)
		tmp = Float64(Float64(t_0 * J) + U);
	else
		tmp = Float64(U + cbrt((Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -110.0], N[(U + N[(t$95$0 * N[(J + N[(K * N[(-0.125 * N[(J * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.1e-10], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+153], N[(N[(t$95$0 * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[Power[N[Power[N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -110

   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 80.3%

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

      5. unpow2 80.3%

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

      6. associate-*r* 80.3%

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

      7. associate-*r* 80.3%

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

   4. Simplified 80.3%

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

   if -110 < l < 2.1e-10

   1. Initial program 67.7%

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

   2. Taylor expanded in l around 0 99.9%

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

   if 2.1e-10 < l < 6.49999999999999972e153

   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 86.4%

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

   if 6.49999999999999972e153 < 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 68.2%

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

      1. add-cbrt-cube 91.8%

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

      2. pow3 91.8%

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

      3. associate-*l* 91.8%

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

      4. associate-*l* 91.8%

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

      5. div-inv 91.8%

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

      6. metadata-eval 91.8%

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

   4. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)}^{3}}} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -110:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + K \cdot \left(-0.125 \cdot \left(J \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \sqrt[3]{{\left(J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)}^{3}}\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.015:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.015)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.015) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.015d0)) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.015) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.015:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.015)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.015)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.015], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.014999999999999999

   1. Initial program 81.1%

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

   2. Taylor expanded in l around 0 66.0%

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

   if -0.014999999999999999 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.7%

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

   2. Taylor expanded in K around 0 85.7%

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

   3. Taylor expanded in l around 0 84.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 7: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ t_1 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ t_2 := U + t_1\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+229}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))
        (t_1 (* 0.3333333333333333 (* J (pow l 3.0))))
        (t_2 (+ U t_1)))
   (if (<= l -2e+42)
     t_2
     (if (<= l 2.1e-10)
       t_0
       (if (<= l 5e+229) t_2 (if (<= l 3e+285) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	double t_1 = 0.3333333333333333 * (J * pow(l, 3.0));
	double t_2 = U + t_1;
	double tmp;
	if (l <= -2e+42) {
		tmp = t_2;
	} else if (l <= 2.1e-10) {
		tmp = t_0;
	} else if (l <= 5e+229) {
		tmp = t_2;
	} else if (l <= 3e+285) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    t_1 = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    t_2 = u + t_1
    if (l <= (-2d+42)) then
        tmp = t_2
    else if (l <= 2.1d-10) then
        tmp = t_0
    else if (l <= 5d+229) then
        tmp = t_2
    else if (l <= 3d+285) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	double t_1 = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	double t_2 = U + t_1;
	double tmp;
	if (l <= -2e+42) {
		tmp = t_2;
	} else if (l <= 2.1e-10) {
		tmp = t_0;
	} else if (l <= 5e+229) {
		tmp = t_2;
	} else if (l <= 3e+285) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	t_1 = 0.3333333333333333 * (J * math.pow(l, 3.0))
	t_2 = U + t_1
	tmp = 0
	if l <= -2e+42:
		tmp = t_2
	elif l <= 2.1e-10:
		tmp = t_0
	elif l <= 5e+229:
		tmp = t_2
	elif l <= 3e+285:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))))
	t_1 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	t_2 = Float64(U + t_1)
	tmp = 0.0
	if (l <= -2e+42)
		tmp = t_2;
	elseif (l <= 2.1e-10)
		tmp = t_0;
	elseif (l <= 5e+229)
		tmp = t_2;
	elseif (l <= 3e+285)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	t_1 = 0.3333333333333333 * (J * (l ^ 3.0));
	t_2 = U + t_1;
	tmp = 0.0;
	if (l <= -2e+42)
		tmp = t_2;
	elseif (l <= 2.1e-10)
		tmp = t_0;
	elseif (l <= 5e+229)
		tmp = t_2;
	elseif (l <= 3e+285)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + t$95$1), $MachinePrecision]}, If[LessEqual[l, -2e+42], t$95$2, If[LessEqual[l, 2.1e-10], t$95$0, If[LessEqual[l, 5e+229], t$95$2, If[LessEqual[l, 3e+285], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.00000000000000009e42 or 2.1e-10 < l < 5.0000000000000005e229

   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 80.5%

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

   3. Taylor expanded in l around 0 58.6%

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

   4. Taylor expanded in l around inf 58.6%

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

   if -2.00000000000000009e42 < l < 2.1e-10 or 5.0000000000000005e229 < l < 3.0000000000000002e285

   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 l around 0 97.8%

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

   if 3.0000000000000002e285 < 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 0 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in J around inf 100.0%

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

   5. Taylor expanded in l around inf 100.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+42}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+229}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+285}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 8: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ t_1 := U + t_0\\ t_2 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+284}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (pow l 3.0))))
        (t_1 (+ U t_0))
        (t_2 (cos (* K 0.5))))
   (if (<= l -1.8e+42)
     t_1
     (if (<= l 2.1e-10)
       (+ U (* (* l 2.0) (* J t_2)))
       (if (<= l 5e+229)
         t_1
         (if (<= l 9.5e+284) (+ U (* 2.0 (* J (* l t_2)))) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * pow(l, 3.0));
	double t_1 = U + t_0;
	double t_2 = cos((K * 0.5));
	double tmp;
	if (l <= -1.8e+42) {
		tmp = t_1;
	} else if (l <= 2.1e-10) {
		tmp = U + ((l * 2.0) * (J * t_2));
	} else if (l <= 5e+229) {
		tmp = t_1;
	} else if (l <= 9.5e+284) {
		tmp = U + (2.0 * (J * (l * t_2)));
	} else {
		tmp = t_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) :: t_2
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    t_1 = u + t_0
    t_2 = cos((k * 0.5d0))
    if (l <= (-1.8d+42)) then
        tmp = t_1
    else if (l <= 2.1d-10) then
        tmp = u + ((l * 2.0d0) * (j * t_2))
    else if (l <= 5d+229) then
        tmp = t_1
    else if (l <= 9.5d+284) then
        tmp = u + (2.0d0 * (j * (l * t_2)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	double t_1 = U + t_0;
	double t_2 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -1.8e+42) {
		tmp = t_1;
	} else if (l <= 2.1e-10) {
		tmp = U + ((l * 2.0) * (J * t_2));
	} else if (l <= 5e+229) {
		tmp = t_1;
	} else if (l <= 9.5e+284) {
		tmp = U + (2.0 * (J * (l * t_2)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * (J * math.pow(l, 3.0))
	t_1 = U + t_0
	t_2 = math.cos((K * 0.5))
	tmp = 0
	if l <= -1.8e+42:
		tmp = t_1
	elif l <= 2.1e-10:
		tmp = U + ((l * 2.0) * (J * t_2))
	elif l <= 5e+229:
		tmp = t_1
	elif l <= 9.5e+284:
		tmp = U + (2.0 * (J * (l * t_2)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	t_1 = Float64(U + t_0)
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -1.8e+42)
		tmp = t_1;
	elseif (l <= 2.1e-10)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * t_2)));
	elseif (l <= 5e+229)
		tmp = t_1;
	elseif (l <= 9.5e+284)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_2))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (J * (l ^ 3.0));
	t_1 = U + t_0;
	t_2 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -1.8e+42)
		tmp = t_1;
	elseif (l <= 2.1e-10)
		tmp = U + ((l * 2.0) * (J * t_2));
	elseif (l <= 5e+229)
		tmp = t_1;
	elseif (l <= 9.5e+284)
		tmp = U + (2.0 * (J * (l * t_2)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.8e+42], t$95$1, If[LessEqual[l, 2.1e-10], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+229], t$95$1, If[LessEqual[l, 9.5e+284], N[(U + N[(2.0 * N[(J * N[(l * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8e42 or 2.1e-10 < l < 5.0000000000000005e229

   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 80.5%

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

   3. Taylor expanded in l around 0 58.6%

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

   4. Taylor expanded in l around inf 58.6%

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

   if -1.8e42 < l < 2.1e-10

   1. Initial program 68.2%

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

   2. Taylor expanded in l around 0 98.3%

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

      1. associate-*r* 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*l* 98.3%

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

      4. associate-*r* 98.3%

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

   4. Simplified 98.3%

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

   if 5.0000000000000005e229 < l < 9.4999999999999997e284

   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 94.8%

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

   if 9.4999999999999997e284 < 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 0 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in J around inf 100.0%

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

   5. Taylor expanded in l around inf 100.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+42}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+229}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+284}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 9: 76.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ t_1 := U + t_0\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+284}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* J (pow l 3.0)))) (t_1 (+ U t_0)))
   (if (<= l -2e+42)
     t_1
     (if (<= l 2.1e-10)
       (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
       (if (<= l 1.6e+229)
         t_1
         (if (<= l 2.35e+284)
           (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
           t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * pow(l, 3.0));
	double t_1 = U + t_0;
	double tmp;
	if (l <= -2e+42) {
		tmp = t_1;
	} else if (l <= 2.1e-10) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.6e+229) {
		tmp = t_1;
	} else if (l <= 2.35e+284) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = t_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 = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    t_1 = u + t_0
    if (l <= (-2d+42)) then
        tmp = t_1
    else if (l <= 2.1d-10) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if (l <= 1.6d+229) then
        tmp = t_1
    else if (l <= 2.35d+284) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	double t_1 = U + t_0;
	double tmp;
	if (l <= -2e+42) {
		tmp = t_1;
	} else if (l <= 2.1e-10) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.6e+229) {
		tmp = t_1;
	} else if (l <= 2.35e+284) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * (J * math.pow(l, 3.0))
	t_1 = U + t_0
	tmp = 0
	if l <= -2e+42:
		tmp = t_1
	elif l <= 2.1e-10:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif l <= 1.6e+229:
		tmp = t_1
	elif l <= 2.35e+284:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)))
	t_1 = Float64(U + t_0)
	tmp = 0.0
	if (l <= -2e+42)
		tmp = t_1;
	elseif (l <= 2.1e-10)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 1.6e+229)
		tmp = t_1;
	elseif (l <= 2.35e+284)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (J * (l ^ 3.0));
	t_1 = U + t_0;
	tmp = 0.0;
	if (l <= -2e+42)
		tmp = t_1;
	elseif (l <= 2.1e-10)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif (l <= 1.6e+229)
		tmp = t_1;
	elseif (l <= 2.35e+284)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + t$95$0), $MachinePrecision]}, If[LessEqual[l, -2e+42], t$95$1, If[LessEqual[l, 2.1e-10], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+229], t$95$1, If[LessEqual[l, 2.35e+284], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.00000000000000009e42 or 2.1e-10 < l < 1.5999999999999999e229

   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 80.5%

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

   3. Taylor expanded in l around 0 58.6%

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

   4. Taylor expanded in l around inf 58.6%

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

   if -2.00000000000000009e42 < l < 2.1e-10

   1. Initial program 68.2%

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

   2. Taylor expanded in l around 0 98.3%

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

   if 1.5999999999999999e229 < l < 2.35000000000000011e284

   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 94.8%

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

   if 2.35000000000000011e284 < 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 0 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in J around inf 100.0%

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

   5. Taylor expanded in l around inf 100.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+42}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+229}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+284}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 10: 72.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+42} \lor \neg \left(\ell \leq 2.1 \cdot 10^{-10}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.8e+42) (not (<= l 2.1e-10)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (fma (* J 2.0) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.8e+42) || !(l <= 2.1e-10)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = fma((J * 2.0), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.8e+42) || !(l <= 2.1e-10))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = fma(Float64(J * 2.0), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.8e+42], N[Not[LessEqual[l, 2.1e-10]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8e42 or 2.1e-10 < 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%

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

   3. Taylor expanded in l around 0 61.6%

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

   4. Taylor expanded in l around inf 61.6%

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

   if -1.8e42 < l < 2.1e-10

   1. Initial program 68.2%

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

   2. Taylor expanded in K around 0 66.6%

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

   3. Taylor expanded in l around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-*r* 83.6%

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

      3. fma-def 83.6%

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

      4. *-commutative 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+42} \lor \neg \left(\ell \leq 2.1 \cdot 10^{-10}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \]

Alternative 11: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+42} \lor \neg \left(\ell \leq 9.8 \cdot 10^{+44}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.8e+42) (not (<= l 9.8e+44)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (fma (* J 2.0) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.8e+42) || !(l <= 9.8e+44)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else {
		tmp = fma((J * 2.0), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.8e+42) || !(l <= 9.8e+44))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	else
		tmp = fma(Float64(J * 2.0), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.8e+42], N[Not[LessEqual[l, 9.8e+44]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.8e42 or 9.80000000000000071e44 < 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 78.3%

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

   3. Taylor expanded in l around 0 66.5%

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

   4. Taylor expanded in J around inf 66.4%

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

   5. Taylor expanded in l around inf 66.4%

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

   if -1.8e42 < l < 9.80000000000000071e44

   1. Initial program 71.0%

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

   2. Taylor expanded in K around 0 68.8%

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

   3. Taylor expanded in l around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. associate-*r* 77.3%

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

      3. fma-def 77.3%

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

      4. *-commutative 77.3%

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

   5. Simplified 77.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+42} \lor \neg \left(\ell \leq 9.8 \cdot 10^{+44}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \]

Alternative 12: 55.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2300:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+168}:\\ \;\;\;\;{U}^{-134217728}\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2300.0)
   (fma (* J 2.0) l U)
   (if (<= l 6.6e+168) (pow U -134217728.0) (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2300.0) {
		tmp = fma((J * 2.0), l, U);
	} else if (l <= 6.6e+168) {
		tmp = pow(U, -134217728.0);
	} else {
		tmp = J * (l * 2.0);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2300.0)
		tmp = fma(Float64(J * 2.0), l, U);
	elseif (l <= 6.6e+168)
		tmp = U ^ -134217728.0;
	else
		tmp = Float64(J * Float64(l * 2.0));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2300.0], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[l, 6.6e+168], N[Power[U, -134217728.0], $MachinePrecision], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2300

   1. Initial program 78.8%

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

   2. Taylor expanded in K around 0 70.7%

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

   3. Taylor expanded in l around 0 64.8%

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

      1. +-commutative 64.8%

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

      2. associate-*r* 64.8%

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

      3. fma-def 64.8%

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

      4. *-commutative 64.8%

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

   5. Simplified 64.8%

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

   if 2300 < l < 6.5999999999999997e168

   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%

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

   4. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{{U}^{-134217728}} \]

   if 6.5999999999999997e168 < 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 80.6%

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

   3. Taylor expanded in l around 0 80.6%

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

   4. Taylor expanded in J around inf 80.6%

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

   5. Taylor expanded in l around 0 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*r* 59.1%

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

      3. *-commutative 59.1%

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

   7. Simplified 59.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2300:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+168}:\\ \;\;\;\;{U}^{-134217728}\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 13: 55.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq 2300:\\ \;\;\;\;U + t_0\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+168}:\\ \;\;\;\;{U}^{-134217728}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l 2.0))))
   (if (<= l 2300.0) (+ U t_0) (if (<= l 1.8e+168) (pow U -134217728.0) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= 2300.0) {
		tmp = U + t_0;
	} else if (l <= 1.8e+168) {
		tmp = pow(U, -134217728.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (l * 2.0d0)
    if (l <= 2300.0d0) then
        tmp = u + t_0
    else if (l <= 1.8d+168) then
        tmp = u ** (-134217728.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= 2300.0) {
		tmp = U + t_0;
	} else if (l <= 1.8e+168) {
		tmp = Math.pow(U, -134217728.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * 2.0)
	tmp = 0
	if l <= 2300.0:
		tmp = U + t_0
	elif l <= 1.8e+168:
		tmp = math.pow(U, -134217728.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * 2.0))
	tmp = 0.0
	if (l <= 2300.0)
		tmp = Float64(U + t_0);
	elseif (l <= 1.8e+168)
		tmp = U ^ -134217728.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * 2.0);
	tmp = 0.0;
	if (l <= 2300.0)
		tmp = U + t_0;
	elseif (l <= 1.8e+168)
		tmp = U ^ -134217728.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2300.0], N[(U + t$95$0), $MachinePrecision], If[LessEqual[l, 1.8e+168], N[Power[U, -134217728.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if l < 2300

   1. Initial program 78.8%

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

   2. Taylor expanded in K around 0 70.7%

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

   3. Taylor expanded in l around 0 64.8%

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

   if 2300 < l < 1.8e168

   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%

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

   4. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{{U}^{-134217728}} \]

   if 1.8e168 < 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 80.6%

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

   3. Taylor expanded in l around 0 80.6%

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

   4. Taylor expanded in J around inf 80.6%

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

   5. Taylor expanded in l around 0 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*r* 59.1%

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

      3. *-commutative 59.1%

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

   7. Simplified 59.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2300:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+168}:\\ \;\;\;\;{U}^{-134217728}\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 14: 45.4% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 215:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.24 \cdot 10^{+179}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l 2.0))))
   (if (<= l -5.2e-34)
     t_0
     (if (<= l 215.0) U (if (<= l 1.24e+179) (* U (- 2.0 U)) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= -5.2e-34) {
		tmp = t_0;
	} else if (l <= 215.0) {
		tmp = U;
	} else if (l <= 1.24e+179) {
		tmp = U * (2.0 - U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (l * 2.0d0)
    if (l <= (-5.2d-34)) then
        tmp = t_0
    else if (l <= 215.0d0) then
        tmp = u
    else if (l <= 1.24d+179) then
        tmp = u * (2.0d0 - u)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= -5.2e-34) {
		tmp = t_0;
	} else if (l <= 215.0) {
		tmp = U;
	} else if (l <= 1.24e+179) {
		tmp = U * (2.0 - U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * 2.0)
	tmp = 0
	if l <= -5.2e-34:
		tmp = t_0
	elif l <= 215.0:
		tmp = U
	elif l <= 1.24e+179:
		tmp = U * (2.0 - U)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * 2.0))
	tmp = 0.0
	if (l <= -5.2e-34)
		tmp = t_0;
	elseif (l <= 215.0)
		tmp = U;
	elseif (l <= 1.24e+179)
		tmp = Float64(U * Float64(2.0 - U));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * 2.0);
	tmp = 0.0;
	if (l <= -5.2e-34)
		tmp = t_0;
	elseif (l <= 215.0)
		tmp = U;
	elseif (l <= 1.24e+179)
		tmp = U * (2.0 - U);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.2e-34], t$95$0, If[LessEqual[l, 215.0], U, If[LessEqual[l, 1.24e+179], N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.1999999999999999e-34 or 1.23999999999999996e179 < l

   1. Initial program 96.0%

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

   2. Taylor expanded in K around 0 74.1%

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

   3. Taylor expanded in l around 0 69.5%

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

   4. Taylor expanded in J around inf 68.3%

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

   5. Taylor expanded in l around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*r* 37.3%

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

      3. *-commutative 37.3%

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

   7. Simplified 37.3%

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

   if -5.1999999999999999e-34 < l < 215

   1. Initial program 69.8%

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

   2. Taylor expanded in J around 0 69.5%

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

   if 215 < l < 1.23999999999999996e179

   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 10.5%

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

   5. Applied egg-rr 27.7%

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

      1. fma-udef 27.7%

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

      2. +-commutative 27.7%

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

      3. associate-+r+ 27.7%

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

      4. count-2 27.7%

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

      5. distribute-rgt-out 27.7%

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

      6. unsub-neg 27.7%

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

   7. Simplified 27.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-34}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 215:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 1.24 \cdot 10^{+179}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 15: 45.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-34} \lor \neg \left(\ell \leq 1.2 \cdot 10^{+45}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.2e-34) (not (<= l 1.2e+45))) (* J (* l 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.2e-34) || !(l <= 1.2e+45)) {
		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 ((l <= (-5.2d-34)) .or. (.not. (l <= 1.2d+45))) 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 ((l <= -5.2e-34) || !(l <= 1.2e+45)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.2e-34) or not (l <= 1.2e+45):
		tmp = J * (l * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.2e-34) || !(l <= 1.2e+45))
		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 ((l <= -5.2e-34) || ~((l <= 1.2e+45)))
		tmp = J * (l * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.2e-34], N[Not[LessEqual[l, 1.2e+45]], $MachinePrecision]], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.1999999999999999e-34 or 1.19999999999999995e45 < l

   1. Initial program 97.0%

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

   2. Taylor expanded in K around 0 74.9%

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

   3. Taylor expanded in l around 0 65.3%

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

   4. Taylor expanded in J around inf 64.4%

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

   5. Taylor expanded in l around 0 30.5%

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

      1. *-commutative 30.5%

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

      2. associate-*r* 30.5%

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

      3. *-commutative 30.5%

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

   7. Simplified 30.5%

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

   if -5.1999999999999999e-34 < l < 1.19999999999999995e45

   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 63.8%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-34} \lor \neg \left(\ell \leq 1.2 \cdot 10^{+45}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 16: 54.2% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq 3100000:\\ \;\;\;\;U + t_0\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{+174}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l 2.0))))
   (if (<= l 3100000.0) (+ U t_0) (if (<= l 1.62e+174) (* U (- 2.0 U)) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= 3100000.0) {
		tmp = U + t_0;
	} else if (l <= 1.62e+174) {
		tmp = U * (2.0 - U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (l * 2.0d0)
    if (l <= 3100000.0d0) then
        tmp = u + t_0
    else if (l <= 1.62d+174) then
        tmp = u * (2.0d0 - u)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (l * 2.0);
	double tmp;
	if (l <= 3100000.0) {
		tmp = U + t_0;
	} else if (l <= 1.62e+174) {
		tmp = U * (2.0 - U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * 2.0)
	tmp = 0
	if l <= 3100000.0:
		tmp = U + t_0
	elif l <= 1.62e+174:
		tmp = U * (2.0 - U)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * 2.0))
	tmp = 0.0
	if (l <= 3100000.0)
		tmp = Float64(U + t_0);
	elseif (l <= 1.62e+174)
		tmp = Float64(U * Float64(2.0 - U));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * 2.0);
	tmp = 0.0;
	if (l <= 3100000.0)
		tmp = U + t_0;
	elseif (l <= 1.62e+174)
		tmp = U * (2.0 - U);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3100000.0], N[(U + t$95$0), $MachinePrecision], If[LessEqual[l, 1.62e+174], N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.1e6

   1. Initial program 78.9%

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

   2. Taylor expanded in K around 0 70.9%

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

   3. Taylor expanded in l around 0 64.4%

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

   if 3.1e6 < l < 1.62000000000000007e174

   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 11.2%

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

   5. Applied egg-rr 29.7%

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

      1. fma-udef 29.7%

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

      2. +-commutative 29.7%

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

      3. associate-+r+ 29.7%

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

      4. count-2 29.7%

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

      5. distribute-rgt-out 29.7%

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

      6. unsub-neg 29.7%

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

   7. Simplified 29.7%

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

   if 1.62000000000000007e174 < 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 80.0%

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

   3. Taylor expanded in l around 0 80.0%

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

   4. Taylor expanded in J around inf 80.0%

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

   5. Taylor expanded in l around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-*r* 60.9%

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

      3. *-commutative 60.9%

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

   7. Simplified 60.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3100000:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{+174}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 17: 42.2% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -480:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 18500000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -480.0) (* U U) (if (<= l 18500000000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -480.0) {
		tmp = U * U;
	} else if (l <= 18500000000.0) {
		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 <= (-480.0d0)) then
        tmp = u * u
    else if (l <= 18500000000.0d0) 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 <= -480.0) {
		tmp = U * U;
	} else if (l <= 18500000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -480.0:
		tmp = U * U
	elif l <= 18500000000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -480.0)
		tmp = Float64(U * U);
	elseif (l <= 18500000000.0)
		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 <= -480.0)
		tmp = U * U;
	elseif (l <= 18500000000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -480.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 18500000000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -480 or 1.85e10 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 14.4%

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

   if -480 < l < 1.85e10

   1. Initial program 68.9%

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

   2. Taylor expanded in J around 0 65.5%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -480:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 18500000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 18: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -3.725290312339702 \cdot 10^{-9} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -3.725290312339702e-9)

C

double code(double J, double l, double K, double U) {
	return -3.725290312339702e-9;
}

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 = -3.725290312339702d-9
end function

Java

public static double code(double J, double l, double K, double U) {
	return -3.725290312339702e-9;
}

Python

def code(J, l, K, U):
	return -3.725290312339702e-9

Julia

function code(J, l, K, U)
	return -3.725290312339702e-9
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -3.725290312339702e-9;
end

Wolfram

code[J_, l_, K_, U_] := -3.725290312339702e-9

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 73.3%

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

4. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{\frac{U}{U + \left(-134217728 \cdot U + -134217728 \cdot U\right)}} \]
5. Step-by-step derivation

   1. associate-+r+ 2.9%

      \[\leadsto \frac{U}{\color{blue}{\left(U + -134217728 \cdot U\right) + -134217728 \cdot U}} \]

   2. distribute-rgt1-in 2.9%

      \[\leadsto \frac{U}{\color{blue}{\left(-134217728 + 1\right) \cdot U} + -134217728 \cdot U} \]

   3. distribute-rgt-out 2.9%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(\left(-134217728 + 1\right) + -134217728\right)}} \]

   4. associate-/r* 2.9%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{\left(-134217728 + 1\right) + -134217728}} \]

   5. *-inverses 2.9%

      \[\leadsto \frac{\color{blue}{1}}{\left(-134217728 + 1\right) + -134217728} \]

   6. metadata-eval 2.9%

      \[\leadsto \frac{1}{\color{blue}{-134217727} + -134217728} \]

   7. metadata-eval 2.9%

      \[\leadsto \frac{1}{\color{blue}{-268435455}} \]

   8. metadata-eval 2.9%

      \[\leadsto \color{blue}{-3.725290312339702 \cdot 10^{-9}} \]

6. Simplified 2.9%

   \[\leadsto \color{blue}{-3.725290312339702 \cdot 10^{-9}} \]

7. Final simplification 2.9%

   \[\leadsto -3.725290312339702 \cdot 10^{-9} \]

Alternative 19: 36.5% 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 33.8%

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

3. Final simplification 33.8%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023293 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))