Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.6%

Time: 13.3s

Alternatives: 18

Speedup: 1.0×

• 49.7% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* 2.0 (* J (log1p (expm1 (* l (cos (* K 0.5))))))) U))

C

double code(double J, double l, double K, double U) {
	return (2.0 * (J * log1p(expm1((l * cos((K * 0.5))))))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (2.0 * (J * Math.log1p(Math.expm1((l * Math.cos((K * 0.5))))))) + U;
}

Python

def code(J, l, K, U):
	return (2.0 * (J * math.log1p(math.expm1((l * math.cos((K * 0.5))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(2.0 * Float64(J * log1p(expm1(Float64(l * cos(Float64(K * 0.5))))))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * N[(J * N[Log[1 + N[(Exp[N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in l around 0 67.5%

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

   1. log1p-expm1-u 99.3%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   2. *-commutative 99.3%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

5. Applied egg-rr 99.3%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]

6. Final simplification 99.3%

   \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \]
7. Add Preprocessing

Alternative 2: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.98:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.26:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.98)
     (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0)))))))
     (if (<= t_0 0.26)
       (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
       (+ U (* 2.0 (* J (log1p (expm1 l)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.98) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	} else if (t_0 <= 0.26) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else {
		tmp = U + (2.0 * (J * log1p(expm1(l))));
	}
	return tmp;
}

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 <= -0.98) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0))))));
	} else if (t_0 <= 0.26) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (2.0 * (J * Math.log1p(Math.expm1(l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.98:
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	elif t_0 <= 0.26:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	else:
		tmp = U + (2.0 * (J * math.log1p(math.expm1(l))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.98)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))));
	elseif (t_0 <= 0.26)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * log1p(expm1(l)))));
	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, -0.98], N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.26], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.97999999999999998

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0 32.0%

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

   4. Taylor expanded in K around 0 76.2%

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

   if -0.97999999999999998 < (cos.f64 (/.f64 K 2)) < 0.26000000000000001

   1. Initial program 82.8%

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

   3. Taylor expanded in l around 0 77.4%

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

      1. associate-*r* 77.3%

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

      2. *-commutative 77.3%

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

      3. associate-*l* 77.4%

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

   5. Simplified 77.4%

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

   if 0.26000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0 65.7%

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

      1. log1p-expm1-u 99.3%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

      2. *-commutative 99.3%

         \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

   5. Applied egg-rr 99.3%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 84.9%

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

      1. expm1-define 96.6%

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

   8. Simplified 96.6%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\ell\right)}\right)\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.98:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.26:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.26:\\ \;\;\;\;U + t\_0 \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.26)
     (+ U (* t_0 (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* J l)))))
     (+ U (* 2.0 (* J (log1p (expm1 l))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.26) {
		tmp = U + (t_0 * ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (J * l))));
	} else {
		tmp = U + (2.0 * (J * log1p(expm1(l))));
	}
	return tmp;
}

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 <= 0.26) {
		tmp = U + (t_0 * ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (J * l))));
	} else {
		tmp = U + (2.0 * (J * Math.log1p(Math.expm1(l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.26:
		tmp = U + (t_0 * ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (J * l))))
	else:
		tmp = U + (2.0 * (J * math.log1p(math.expm1(l))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.26)
		tmp = Float64(U + Float64(t_0 * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(J * l)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * log1p(expm1(l)))));
	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, 0.26], N[(U + N[(t$95$0 * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.26000000000000001

   1. Initial program 84.0%

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

   3. Taylor expanded in l around 0 89.5%

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

   if 0.26000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0 65.7%

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

      1. log1p-expm1-u 99.3%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

      2. *-commutative 99.3%

         \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

   5. Applied egg-rr 99.3%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 84.9%

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

      1. expm1-define 96.6%

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

   8. Simplified 96.6%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\ell\right)}\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.26:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.26:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.26)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* 2.0 l)))))
     (+ U (* 2.0 (* J (log1p (expm1 l))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.26) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (2.0 * l))));
	} else {
		tmp = U + (2.0 * (J * log1p(expm1(l))));
	}
	return tmp;
}

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 <= 0.26) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (2.0 * l))));
	} else {
		tmp = U + (2.0 * (J * Math.log1p(Math.expm1(l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.26:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (2.0 * l))))
	else:
		tmp = U + (2.0 * (J * math.log1p(math.expm1(l))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.26)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(2.0 * l)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * log1p(expm1(l)))));
	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, 0.26], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[Log[1 + N[(Exp[l] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.26000000000000001

   1. Initial program 84.0%

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

   3. Taylor expanded in l around 0 89.5%

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

   if 0.26000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0 65.7%

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

      1. log1p-expm1-u 99.3%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

      2. *-commutative 99.3%

         \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

   5. Applied egg-rr 99.3%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 84.9%

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

      1. expm1-define 96.6%

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

   8. Simplified 96.6%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\ell\right)}\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.26:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(U + 4\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 820:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0)))))))))
   (if (<= l -4.1e+92)
     t_0
     (if (<= l -2.45e+67)
       (log1p (expm1 (* U (+ U 4.0))))
       (if (<= l -1.55e+23)
         t_0
         (if (<= l 820.0)
           (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
           (if (<= l 1.65e+131) (log1p (expm1 U)) t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	double tmp;
	if (l <= -4.1e+92) {
		tmp = t_0;
	} else if (l <= -2.45e+67) {
		tmp = log1p(expm1((U * (U + 4.0))));
	} else if (l <= -1.55e+23) {
		tmp = t_0;
	} else if (l <= 820.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 1.65e+131) {
		tmp = log1p(expm1(U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0))))));
	double tmp;
	if (l <= -4.1e+92) {
		tmp = t_0;
	} else if (l <= -2.45e+67) {
		tmp = Math.log1p(Math.expm1((U * (U + 4.0))));
	} else if (l <= -1.55e+23) {
		tmp = t_0;
	} else if (l <= 820.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 1.65e+131) {
		tmp = Math.log1p(Math.expm1(U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	tmp = 0
	if l <= -4.1e+92:
		tmp = t_0
	elif l <= -2.45e+67:
		tmp = math.log1p(math.expm1((U * (U + 4.0))))
	elif l <= -1.55e+23:
		tmp = t_0
	elif l <= 820.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 1.65e+131:
		tmp = math.log1p(math.expm1(U))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))))
	tmp = 0.0
	if (l <= -4.1e+92)
		tmp = t_0;
	elseif (l <= -2.45e+67)
		tmp = log1p(expm1(Float64(U * Float64(U + 4.0))));
	elseif (l <= -1.55e+23)
		tmp = t_0;
	elseif (l <= 820.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 1.65e+131)
		tmp = log1p(expm1(U));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.1e+92], t$95$0, If[LessEqual[l, -2.45e+67], N[Log[1 + N[(Exp[N[(U * N[(U + 4.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -1.55e+23], t$95$0, If[LessEqual[l, 820.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+131], N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.10000000000000024e92 or -2.44999999999999995e67 < l < -1.54999999999999985e23 or 1.6499999999999999e131 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 34.6%

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

   4. Taylor expanded in K around 0 52.3%

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

   if -4.10000000000000024e92 < l < -2.44999999999999995e67

   1. Initial program 100.0%

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

   4. Applied egg-rr 28.2%

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

      1. log1p-expm1-u 87.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(U - -4\right)\right)\right)} \]

      2. sub-neg 87.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \color{blue}{\left(U + \left(--4\right)\right)}\right)\right) \]

      3. metadata-eval 87.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(U + \color{blue}{4}\right)\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(U + 4\right)\right)\right)} \]

   if -1.54999999999999985e23 < l < 820

   1. Initial program 73.8%

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

   3. Taylor expanded in l around 0 96.9%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*l* 96.9%

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

   5. Simplified 96.9%

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

   if 820 < l < 1.6499999999999999e131

   1. Initial program 100.0%

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

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+92}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U \cdot \left(U + 4\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 820:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 940:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0)))))))))
   (if (<= l -1.55e+23)
     t_0
     (if (<= l 940.0)
       (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
       (if (<= l 1.35e+131) (log1p (expm1 U)) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	double tmp;
	if (l <= -1.55e+23) {
		tmp = t_0;
	} else if (l <= 940.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 1.35e+131) {
		tmp = log1p(expm1(U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0))))));
	double tmp;
	if (l <= -1.55e+23) {
		tmp = t_0;
	} else if (l <= 940.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 1.35e+131) {
		tmp = Math.log1p(Math.expm1(U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	tmp = 0
	if l <= -1.55e+23:
		tmp = t_0
	elif l <= 940.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 1.35e+131:
		tmp = math.log1p(math.expm1(U))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))))
	tmp = 0.0
	if (l <= -1.55e+23)
		tmp = t_0;
	elseif (l <= 940.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 1.35e+131)
		tmp = log1p(expm1(U));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.55e+23], t$95$0, If[LessEqual[l, 940.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e+131], N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.54999999999999985e23 or 1.35000000000000002e131 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 32.9%

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

   4. Taylor expanded in K around 0 50.1%

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

   if -1.54999999999999985e23 < l < 940

   1. Initial program 73.8%

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

   3. Taylor expanded in l around 0 96.9%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*l* 96.9%

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

   5. Simplified 96.9%

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

   if 940 < l < 1.35000000000000002e131

   1. Initial program 100.0%

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

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 940:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+130}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot t\_1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+247}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0))))))))
        (t_1 (cos (* K 0.5))))
   (if (<= l -1.55e+23)
     t_0
     (if (<= l 8.2e+130)
       (+ U (* 2.0 (* l (* J t_1))))
       (if (<= l 1.75e+247) t_0 (+ U (* 2.0 (* J (* l t_1)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -1.55e+23) {
		tmp = t_0;
	} else if (l <= 8.2e+130) {
		tmp = U + (2.0 * (l * (J * t_1)));
	} else if (l <= 1.75e+247) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k ** 2.0d0))))))
    t_1 = cos((k * 0.5d0))
    if (l <= (-1.55d+23)) then
        tmp = t_0
    else if (l <= 8.2d+130) then
        tmp = u + (2.0d0 * (l * (j * t_1)))
    else if (l <= 1.75d+247) then
        tmp = t_0
    else
        tmp = u + (2.0d0 * (j * (l * 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 + (-0.125 * (l * Math.pow(K, 2.0))))));
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -1.55e+23) {
		tmp = t_0;
	} else if (l <= 8.2e+130) {
		tmp = U + (2.0 * (l * (J * t_1)));
	} else if (l <= 1.75e+247) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * t_1)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -1.55e+23:
		tmp = t_0
	elif l <= 8.2e+130:
		tmp = U + (2.0 * (l * (J * t_1)))
	elif l <= 1.75e+247:
		tmp = t_0
	else:
		tmp = U + (2.0 * (J * (l * t_1)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))))
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -1.55e+23)
		tmp = t_0;
	elseif (l <= 8.2e+130)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * t_1))));
	elseif (l <= 1.75e+247)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (J * (l + (-0.125 * (l * (K ^ 2.0))))));
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -1.55e+23)
		tmp = t_0;
	elseif (l <= 8.2e+130)
		tmp = U + (2.0 * (l * (J * t_1)));
	elseif (l <= 1.75e+247)
		tmp = t_0;
	else
		tmp = U + (2.0 * (J * (l * 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[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.55e+23], t$95$0, If[LessEqual[l, 8.2e+130], N[(U + N[(2.0 * N[(l * N[(J * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+247], t$95$0, N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.54999999999999985e23 or 8.19999999999999955e130 < l < 1.7500000000000001e247

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 26.9%

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

   4. Taylor expanded in K around 0 49.0%

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

   if -1.54999999999999985e23 < l < 8.19999999999999955e130

   1. Initial program 77.3%

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

   3. Taylor expanded in l around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-*l* 85.3%

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

   5. Simplified 85.3%

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

   if 1.7500000000000001e247 < 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. Add Preprocessing

   3. Taylor expanded in l around 0 61.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+130}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+247}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.3 \cdot 10^{+244}:\\ \;\;\;\;\frac{U \cdot \mathsf{fma}\left(U, U, -16\right)}{U + -4}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{+143}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+19}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6.3e+244)
   (/ (* U (fma U U -16.0)) (+ U -4.0))
   (if (<= l -2.4e+143)
     (pow U -3.0)
     (if (<= l -9e+19)
       (+ U (* J (+ -0.5 (* (pow K 2.0) 0.0625))))
       (+ U (* 2.0 (* J l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.3e+244) {
		tmp = (U * fma(U, U, -16.0)) / (U + -4.0);
	} else if (l <= -2.4e+143) {
		tmp = pow(U, -3.0);
	} else if (l <= -9e+19) {
		tmp = U + (J * (-0.5 + (pow(K, 2.0) * 0.0625)));
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6.3e+244)
		tmp = Float64(Float64(U * fma(U, U, -16.0)) / Float64(U + -4.0));
	elseif (l <= -2.4e+143)
		tmp = U ^ -3.0;
	elseif (l <= -9e+19)
		tmp = Float64(U + Float64(J * Float64(-0.5 + Float64((K ^ 2.0) * 0.0625))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6.3e+244], N[(N[(U * N[(U * U + -16.0), $MachinePrecision]), $MachinePrecision] / N[(U + -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.4e+143], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, -9e+19], N[(U + N[(J * N[(-0.5 + N[(N[Power[K, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.2999999999999996e244

   1. Initial program 100.0%

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

   4. Applied egg-rr 33.1%

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

      1. *-commutative 33.1%

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

      2. flip-- 33.1%

         \[\leadsto \color{blue}{\frac{U \cdot U - -4 \cdot -4}{U + -4}} \cdot U \]

      3. associate-*l/ 54.6%

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

      4. fma-neg 54.6%

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

      5. metadata-eval 54.6%

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

      6. metadata-eval 54.6%

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

   6. Applied egg-rr 54.6%

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

   if -6.2999999999999996e244 < l < -2.3999999999999998e143

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.9%

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

   if -2.3999999999999998e143 < l < -9e19

   1. Initial program 100.0%

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

   4. Applied egg-rr 3.7%

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

   5. Taylor expanded in K around 0 32.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot J + 0.0625 \cdot \left(J \cdot {K}^{2}\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 32.4%

         \[\leadsto \left(-0.5 \cdot J + 0.0625 \cdot \color{blue}{\left({K}^{2} \cdot J\right)}\right) + U \]

      2. associate-*r* 32.4%

         \[\leadsto \left(-0.5 \cdot J + \color{blue}{\left(0.0625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      3. distribute-rgt-out 32.4%

         \[\leadsto \color{blue}{J \cdot \left(-0.5 + 0.0625 \cdot {K}^{2}\right)} + U \]

   7. Simplified 32.4%

      \[\leadsto \color{blue}{J \cdot \left(-0.5 + 0.0625 \cdot {K}^{2}\right)} + U \]

   if -9e19 < l

   1. Initial program 81.0%

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

   3. Taylor expanded in l around 0 78.7%

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

   4. Taylor expanded in K around 0 66.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.3 \cdot 10^{+244}:\\ \;\;\;\;\frac{U \cdot \mathsf{fma}\left(U, U, -16\right)}{U + -4}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{+143}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+19}:\\ \;\;\;\;U + J \cdot \left(-0.5 + {K}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{U \cdot \mathsf{fma}\left(U, U, -16\right)}{U + -4}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{+95}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq -950:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (/ (* U (fma U U -16.0)) (+ U -4.0))))
   (if (<= l -7.2e+247)
     t_0
     (if (<= l -2.05e+95)
       (pow U -3.0)
       (if (<= l -950.0) t_0 (+ U (* 2.0 (* J l))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (U * fma(U, U, -16.0)) / (U + -4.0);
	double tmp;
	if (l <= -7.2e+247) {
		tmp = t_0;
	} else if (l <= -2.05e+95) {
		tmp = pow(U, -3.0);
	} else if (l <= -950.0) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(U * fma(U, U, -16.0)) / Float64(U + -4.0))
	tmp = 0.0
	if (l <= -7.2e+247)
		tmp = t_0;
	elseif (l <= -2.05e+95)
		tmp = U ^ -3.0;
	elseif (l <= -950.0)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(U * N[(U * U + -16.0), $MachinePrecision]), $MachinePrecision] / N[(U + -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.2e+247], t$95$0, If[LessEqual[l, -2.05e+95], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, -950.0], t$95$0, N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.2e247 or -2.04999999999999993e95 < l < -950

   1. Initial program 100.0%

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

   4. Applied egg-rr 27.9%

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

      1. *-commutative 27.9%

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

      2. flip-- 27.9%

         \[\leadsto \color{blue}{\frac{U \cdot U - -4 \cdot -4}{U + -4}} \cdot U \]

      3. associate-*l/ 39.8%

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

      4. fma-neg 39.8%

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

      5. metadata-eval 39.8%

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

      6. metadata-eval 39.8%

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

   6. Applied egg-rr 39.8%

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

   if -7.2e247 < l < -2.04999999999999993e95

   1. Initial program 100.0%

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

   4. Applied egg-rr 41.4%

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

   if -950 < l

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0 79.4%

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

   4. Taylor expanded in K around 0 67.0%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+247}:\\ \;\;\;\;\frac{U \cdot \mathsf{fma}\left(U, U, -16\right)}{U + -4}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{+95}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq -950:\\ \;\;\;\;\frac{U \cdot \mathsf{fma}\left(U, U, -16\right)}{U + -4}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+240}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{+103}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.7e+240)
   (* U U)
   (if (<= l -6.2e+103) (pow U -3.0) (+ U (* 2.0 (* J l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.7e+240) {
		tmp = U * U;
	} else if (l <= -6.2e+103) {
		tmp = pow(U, -3.0);
	} else {
		tmp = U + (2.0 * (J * l));
	}
	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 <= (-5.7d+240)) then
        tmp = u * u
    else if (l <= (-6.2d+103)) then
        tmp = u ** (-3.0d0)
    else
        tmp = u + (2.0d0 * (j * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.7e+240) {
		tmp = U * U;
	} else if (l <= -6.2e+103) {
		tmp = Math.pow(U, -3.0);
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.7e+240:
		tmp = U * U
	elif l <= -6.2e+103:
		tmp = math.pow(U, -3.0)
	else:
		tmp = U + (2.0 * (J * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.7e+240)
		tmp = Float64(U * U);
	elseif (l <= -6.2e+103)
		tmp = U ^ -3.0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.7e+240)
		tmp = U * U;
	elseif (l <= -6.2e+103)
		tmp = U ^ -3.0;
	else
		tmp = U + (2.0 * (J * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.7e+240], N[(U * U), $MachinePrecision], If[LessEqual[l, -6.2e+103], N[Power[U, -3.0], $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.69999999999999973e240

   1. Initial program 100.0%

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

   4. Applied egg-rr 33.1%

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

   if -5.69999999999999973e240 < l < -6.2000000000000003e103

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.0%

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

   if -6.2000000000000003e103 < l

   1. Initial program 82.5%

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

   3. Taylor expanded in l around 0 73.8%

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

   4. Taylor expanded in K around 0 62.4%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+240}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{+103}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in l around 0 67.5%

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

4. Final simplification 67.5%

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

Alternative 12: 64.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in l around 0 67.5%

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

   1. associate-*r* 67.5%

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

   2. *-commutative 67.5%

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

   3. associate-*l* 67.5%

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

5. Simplified 67.5%

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

6. Final simplification 67.5%

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

Alternative 13: 52.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+104}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.3e+104) (pow U -4.0) (+ U (* 2.0 (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+104) {
		tmp = pow(U, -4.0);
	} else {
		tmp = U + (2.0 * (J * l));
	}
	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 <= (-3.3d+104)) then
        tmp = u ** (-4.0d0)
    else
        tmp = u + (2.0d0 * (j * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+104) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.3e+104:
		tmp = math.pow(U, -4.0)
	else:
		tmp = U + (2.0 * (J * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.3e+104)
		tmp = U ^ -4.0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.3e+104)
		tmp = U ^ -4.0;
	else
		tmp = U + (2.0 * (J * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.3e+104], N[Power[U, -4.0], $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.29999999999999985e104

   1. Initial program 100.0%

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

   4. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -3.29999999999999985e104 < l

   1. Initial program 82.5%

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

   3. Taylor expanded in l around 0 73.8%

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

   4. Taylor expanded in K around 0 62.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+104}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.6% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1000.0) (not (<= l 7.5e-5))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1000.0) || !(l <= 7.5e-5)) {
		tmp = U * (U - -4.0);
	} 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 <= (-1000.0d0)) .or. (.not. (l <= 7.5d-5))) then
        tmp = u * (u - (-4.0d0))
    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 <= -1000.0) || !(l <= 7.5e-5)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1000.0) or not (l <= 7.5e-5):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1000.0) || !(l <= 7.5e-5))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1000.0) || ~((l <= 7.5e-5)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1000.0], N[Not[LessEqual[l, 7.5e-5]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1e3 or 7.49999999999999934e-5 < l

   1. Initial program 99.7%

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

   4. Applied egg-rr 17.6%

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

   if -1e3 < l < 7.49999999999999934e-5

   1. Initial program 72.9%

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

   3. Taylor expanded in J around 0 72.6%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.5% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1250 \lor \neg \left(\ell \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1250.0) (not (<= l 7.5e-5))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1250.0) || !(l <= 7.5e-5)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1250.0d0)) .or. (.not. (l <= 7.5d-5))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1250.0) || !(l <= 7.5e-5)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1250.0) or not (l <= 7.5e-5):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1250.0) || !(l <= 7.5e-5))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1250.0) || ~((l <= 7.5e-5)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1250.0], N[Not[LessEqual[l, 7.5e-5]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1250 or 7.49999999999999934e-5 < l

   1. Initial program 99.7%

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

   4. Applied egg-rr 17.4%

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

   if -1250 < l < 7.49999999999999934e-5

   1. Initial program 72.9%

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

   3. Taylor expanded in J around 0 72.6%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1250 \lor \neg \left(\ell \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 + (2.0d0 * (j * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in l around 0 67.5%

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

4. Taylor expanded in K around 0 55.7%

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

5. Final simplification 55.7%

   \[\leadsto U + 2 \cdot \left(J \cdot \ell\right) \]
6. Add Preprocessing

Alternative 17: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

4. Applied egg-rr 2.9%

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

   1. *-inverses 2.9%

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

6. Simplified 2.9%

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

7. Final simplification 2.9%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 18: 36.7% 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 85.0%

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

3. Taylor expanded in J around 0 40.8%

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

4. Final simplification 40.8%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))