Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.8%

Time: 13.2s

Alternatives: 14

Speedup: 1.4×

• 50.9% 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: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot t_0\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-6)))
     (+ (* (* t_1 J) (cos (/ K 2.0))) U)
     (+
      U
      (+
       (* 0.3333333333333333 (* J (* (pow l 3.0) t_0)))
       (* 2.0 (* J (* l t_0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + ((0.3333333333333333 * (J * (pow(l, 3.0) * t_0))) + (2.0 * (J * (l * t_0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + ((0.3333333333333333 * (J * (Math.pow(l, 3.0) * t_0))) + (2.0 * (J * (l * t_0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-6):
		tmp = ((t_1 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + ((0.3333333333333333 * (J * (math.pow(l, 3.0) * t_0))) + (2.0 * (J * (l * t_0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-6))
		tmp = Float64(Float64(Float64(t_1 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(Float64(0.3333333333333333 * Float64(J * Float64((l ^ 3.0) * t_0))) + Float64(2.0 * Float64(J * Float64(l * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-6)))
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + ((0.3333333333333333 * (J * ((l ^ 3.0) * t_0))) + (2.0 * (J * (l * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $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, 2e-6]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(0.3333333333333333 * N[(J * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 1.99999999999999991e-6 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-6)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (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 <= 2e-6)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-6):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-6))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-6)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-6]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 1.99999999999999991e-6 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.3739:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.3739)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
     (fma (* J 2.0) (sinh l) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.3739) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else {
		tmp = fma((J * 2.0), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.3739)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	else
		tmp = fma(Float64(J * 2.0), sinh(l), U);
	end
	return 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, 0.3739], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.37390000000000001

   1. Initial program 86.3%

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

   2. Taylor expanded in l around 0 88.4%

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

   if 0.37390000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0 86.2%

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

      1. expm1-log1p-u 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\right)\right)} \]

      2. expm1-udef 31.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\right)} - 1} \]

      3. fma-def 31.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)}\right)} - 1 \]

      4. sinh-undef 33.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(J, \color{blue}{2 \cdot \sinh \ell}, U\right)\right)} - 1 \]

   4. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(J, 2 \cdot \sinh \ell, U\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 50.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(J, 2 \cdot \sinh \ell, U\right)\right)\right)} \]

      2. expm1-log1p 96.6%

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

      3. fma-udef 96.6%

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

      4. associate-*r* 96.6%

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

      5. fma-def 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.3739:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.19:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.19)
   (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
   (fma (* J 2.0) (sinh l) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.19) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = fma((J * 2.0), sinh(l), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.19)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = fma(Float64(J * 2.0), sinh(l), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.19], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.19

   1. Initial program 88.0%

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

   2. Taylor expanded in l around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l* 70.7%

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

      3. *-commutative 70.7%

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

      4. associate-*r* 70.7%

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

   4. Simplified 70.7%

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

   if 0.19 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.8%

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

   2. Taylor expanded in K around 0 85.5%

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

      1. expm1-log1p-u 42.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\right)\right)} \]

      2. expm1-udef 30.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\right)} - 1} \]

      3. fma-def 30.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)}\right)} - 1 \]

      4. sinh-undef 32.8%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(J, \color{blue}{2 \cdot \sinh \ell}, U\right)\right)} - 1 \]

   4. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(J, 2 \cdot \sinh \ell, U\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 48.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(J, 2 \cdot \sinh \ell, U\right)\right)\right)} \]

      2. expm1-log1p 95.5%

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

      3. fma-udef 95.5%

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

      4. associate-*r* 95.5%

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

      5. fma-def 95.5%

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

   6. Simplified 95.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.19:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \end{array} \]

Alternative 5: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_1 := \mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.009:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+17}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 3.0) (* J 0.3333333333333333)))))
        (t_1 (fma (* J 2.0) (sinh l) U)))
   (if (<= l -3.7e+93)
     t_0
     (if (<= l -0.009)
       t_1
       (if (<= l 9.2e+17)
         (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
         (if (<= l 3e+73) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 3.0) * (J * 0.3333333333333333)));
	double t_1 = fma((J * 2.0), sinh(l), U);
	double tmp;
	if (l <= -3.7e+93) {
		tmp = t_0;
	} else if (l <= -0.009) {
		tmp = t_1;
	} else if (l <= 9.2e+17) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else if (l <= 3e+73) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))))
	t_1 = fma(Float64(J * 2.0), sinh(l), U)
	tmp = 0.0
	if (l <= -3.7e+93)
		tmp = t_0;
	elseif (l <= -0.009)
		tmp = t_1;
	elseif (l <= 9.2e+17)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	elseif (l <= 3e+73)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * 2.0), $MachinePrecision] * N[Sinh[l], $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -3.7e+93], t$95$0, If[LessEqual[l, -0.009], t$95$1, If[LessEqual[l, 9.2e+17], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3e+73], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.69999999999999987e93 or 3.00000000000000011e73 < 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 96.7%

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

   3. Taylor expanded in l around inf 96.7%

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

      1. associate-*r* 96.7%

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

      2. *-commutative 96.7%

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

   5. Simplified 96.7%

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

   if -3.69999999999999987e93 < l < -0.00899999999999999932 or 9.2e17 < l < 3.00000000000000011e73

   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 84.2%

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

      1. expm1-log1p-u 34.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\right)\right)} \]

      2. expm1-udef 34.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\right)} - 1} \]

      3. fma-def 34.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(J, e^{\ell} - e^{-\ell}, U\right)}\right)} - 1 \]

      4. sinh-undef 34.2%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(J, \color{blue}{2 \cdot \sinh \ell}, U\right)\right)} - 1 \]

   4. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(J, 2 \cdot \sinh \ell, U\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 34.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(J, 2 \cdot \sinh \ell, U\right)\right)\right)} \]

      2. expm1-log1p 84.2%

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

      3. fma-udef 84.2%

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

      4. associate-*r* 84.2%

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

      5. fma-def 84.2%

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

   6. Simplified 84.2%

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

   if -0.00899999999999999932 < l < 9.2e17

   1. Initial program 74.0%

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

   2. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r* 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+93}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.009:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+17}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \sinh \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 6: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.54:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.54)
   (+ U (* (* l 2.0) (* 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)) <= 0.54) {
		tmp = U + ((l * 2.0) * (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)) <= 0.54d0) then
        tmp = u + ((l * 2.0d0) * (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)) <= 0.54) {
		tmp = U + ((l * 2.0) * (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)) <= 0.54:
		tmp = U + ((l * 2.0) * (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)) <= 0.54)
		tmp = Float64(U + Float64(Float64(l * 2.0) * 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)) <= 0.54)
		tmp = U + ((l * 2.0) * (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], 0.54], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.54000000000000004

   1. Initial program 86.2%

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

   2. Taylor expanded in l around 0 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l* 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r* 66.4%

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

   4. Simplified 66.4%

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

   if 0.54000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in l around 0 87.0%

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

   3. Taylor expanded in K around 0 84.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.54:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 7: 77.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+31}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.2e+42)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 4e+31)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.2e+42) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 4e+31) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = 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 (l <= (-1.2d+42)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 4d+31) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = 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 (l <= -1.2e+42) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 4e+31) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.2e+42:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 4e+31:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.2e+42)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 4e+31)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = 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 (l <= -1.2e+42)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 4e+31)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.2e+42], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+31], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1999999999999999e42

   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 88.0%

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

   3. Taylor expanded in K around 0 59.4%

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

   4. Taylor expanded in l around inf 59.5%

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

   if -1.1999999999999999e42 < l < 3.9999999999999999e31

   1. Initial program 76.9%

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

   2. Taylor expanded in l around 0 88.9%

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

   if 3.9999999999999999e31 < 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 76.3%

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

   3. Taylor expanded in K around 0 61.9%

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

   4. Taylor expanded in J around inf 62.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+42}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+31}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 77.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+45}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.9e+45)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 1.5e+29)
     (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
     (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.9e+45) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 1.5e+29) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = 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 (l <= (-1.9d+45)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 1.5d+29) then
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    else
        tmp = 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 (l <= -1.9e+45) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 1.5e+29) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.9e+45:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 1.5e+29:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.9e+45)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 1.5e+29)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = 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 (l <= -1.9e+45)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 1.5e+29)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	else
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.9e+45], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+29], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9000000000000001e45

   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 88.0%

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

   3. Taylor expanded in K around 0 59.4%

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

   4. Taylor expanded in l around inf 59.5%

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

   if -1.9000000000000001e45 < l < 1.5e29

   1. Initial program 76.9%

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

   2. Taylor expanded in l around 0 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*r* 89.0%

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

   4. Simplified 89.0%

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

   if 1.5e29 < 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 76.3%

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

   3. Taylor expanded in K around 0 61.9%

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

   4. Taylor expanded in J around inf 62.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+45}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 9: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -50000:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+29}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -50000.0)
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (if (<= l 3.7e+29)
     (+ U (* J (* l 2.0)))
     (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -50000.0) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else if (l <= 3.7e+29) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = 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 (l <= (-50000.0d0)) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else if (l <= 3.7d+29) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = 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 (l <= -50000.0) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else if (l <= 3.7e+29) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -50000.0:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	elif l <= 3.7e+29:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -50000.0)
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	elseif (l <= 3.7e+29)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = 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 (l <= -50000.0)
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	elseif (l <= 3.7e+29)
		tmp = U + (J * (l * 2.0));
	else
		tmp = J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -50000.0], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e+29], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e4

   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 74.4%

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

   3. Taylor expanded in K around 0 49.4%

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

   4. Taylor expanded in l around inf 49.5%

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

   if -5e4 < l < 3.69999999999999974e29

   1. Initial program 74.9%

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

   2. Taylor expanded in l around 0 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

      3. *-commutative 95.7%

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

      4. associate-*r* 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in K around 0 83.8%

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

   if 3.69999999999999974e29 < 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 76.3%

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

   3. Taylor expanded in K around 0 61.9%

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

   4. Taylor expanded in J around inf 62.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -50000:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+29}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 10: 72.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -116000 \lor \neg \left(\ell \leq 1.35 \cdot 10^{+29}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -116000.0) (not (<= l 1.35e+29)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -116000.0) || !(l <= 1.35e+29)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} 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 <= (-116000.0d0)) .or. (.not. (l <= 1.35d+29))) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    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 <= -116000.0) || !(l <= 1.35e+29)) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -116000.0) or not (l <= 1.35e+29):
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -116000.0) || !(l <= 1.35e+29))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	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 <= -116000.0) || ~((l <= 1.35e+29)))
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -116000.0], N[Not[LessEqual[l, 1.35e+29]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -116000 or 1.35e29 < 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 75.2%

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

   3. Taylor expanded in K around 0 54.6%

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

   4. Taylor expanded in l around inf 54.8%

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

   if -116000 < l < 1.35e29

   1. Initial program 74.9%

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

   2. Taylor expanded in l around 0 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

      3. *-commutative 95.7%

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

      4. associate-*r* 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in K around 0 83.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -116000 \lor \neg \left(\ell \leq 1.35 \cdot 10^{+29}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 11: 59.8% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+35} \lor \neg \left(\ell \leq 320\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + -0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.2e+35) (not (<= l 320.0)))
   (+ U (* J (* l (+ 2.0 (* -0.25 (* K K))))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.2e+35) || !(l <= 320.0)) {
		tmp = U + (J * (l * (2.0 + (-0.25 * (K * K)))));
	} 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 <= (-1.2d+35)) .or. (.not. (l <= 320.0d0))) then
        tmp = u + (j * (l * (2.0d0 + ((-0.25d0) * (k * k)))))
    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 <= -1.2e+35) || !(l <= 320.0)) {
		tmp = U + (J * (l * (2.0 + (-0.25 * (K * K)))));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.2e+35) or not (l <= 320.0):
		tmp = U + (J * (l * (2.0 + (-0.25 * (K * K)))))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.2e+35) || !(l <= 320.0))
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(-0.25 * Float64(K * K))))));
	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 <= -1.2e+35) || ~((l <= 320.0)))
		tmp = U + (J * (l * (2.0 + (-0.25 * (K * K)))));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.2e+35], N[Not[LessEqual[l, 320.0]], $MachinePrecision]], N[(U + N[(J * N[(l * N[(2.0 + N[(-0.25 * N[(K * K), $MachinePrecision]), $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 < -1.20000000000000007e35 or 320 < 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 34.2%

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

      1. *-commutative 34.2%

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

      2. associate-*l* 34.2%

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

      3. *-commutative 34.2%

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

      4. associate-*r* 34.2%

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

   4. Simplified 34.2%

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

   5. Taylor expanded in K around 0 28.3%

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

      1. +-commutative 28.3%

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

      2. fma-def 28.3%

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

      3. *-commutative 28.3%

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

      4. *-commutative 28.3%

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

      5. associate-*l* 28.3%

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

      6. *-commutative 28.3%

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

      7. unpow2 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in J around 0 40.6%

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

      1. unpow2 40.6%

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

      2. associate-*r* 40.6%

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

      3. distribute-rgt-out 40.6%

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

   10. Simplified 40.6%

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

   if -1.20000000000000007e35 < l < 320

   1. Initial program 75.6%

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

   2. Taylor expanded in l around 0 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-*l* 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-*r* 93.0%

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

   4. Simplified 93.0%

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

   5. Taylor expanded in K around 0 81.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+35} \lor \neg \left(\ell \leq 320\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + -0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 12: 43.9% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -750 \lor \neg \left(\ell \leq 2.55 \cdot 10^{-79}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -750.0) (not (<= l 2.55e-79))) (* J (* l 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -750.0) || !(l <= 2.55e-79)) {
		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 <= (-750.0d0)) .or. (.not. (l <= 2.55d-79))) 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 <= -750.0) || !(l <= 2.55e-79)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -750.0) or not (l <= 2.55e-79):
		tmp = J * (l * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -750.0) || !(l <= 2.55e-79))
		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 <= -750.0) || ~((l <= 2.55e-79)))
		tmp = J * (l * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -750.0], N[Not[LessEqual[l, 2.55e-79]], $MachinePrecision]], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -750 or 2.55e-79 < l

   1. Initial program 93.8%

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

   2. Taylor expanded in l around 0 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-*l* 38.9%

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

      3. *-commutative 38.9%

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

      4. associate-*r* 38.9%

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

   4. Simplified 38.9%

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

   5. Taylor expanded in K around 0 27.4%

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

   6. Taylor expanded in l around inf 23.4%

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

      1. *-commutative 23.4%

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

      2. associate-*r* 23.4%

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

   8. Simplified 23.4%

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

   if -750 < l < 2.55e-79

   1. Initial program 77.7%

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

   3. Applied egg-rr 59.6%

      \[\leadsto \left(J \cdot \color{blue}{0.125}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in J around 0 77.7%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -750 \lor \neg \left(\ell \leq 2.55 \cdot 10^{-79}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 54.0% 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 86.4%

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

2. Taylor expanded in l around 0 67.0%

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

   1. *-commutative 67.0%

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

   2. associate-*l* 67.1%

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

   3. *-commutative 67.1%

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

   4. associate-*r* 67.1%

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

4. Simplified 67.1%

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

5. Taylor expanded in K around 0 55.7%

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

6. Final simplification 55.7%

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

Alternative 14: 35.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Applied egg-rr 31.0%

   \[\leadsto \left(J \cdot \color{blue}{0.125}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Taylor expanded in J around 0 39.2%

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

5. Final simplification 39.2%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023287 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))