Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.4%

Time: 16.1s

Alternatives: 16

Speedup: 1.4×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in l around 0 65.9%

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

   1. log1p-expm1-u 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{-9}\right):\\ \;\;\;\;U + J \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-9)))
     (+ U (* J t_0))
     (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e-9)) {
		tmp = U + (J * t_0);
	} else {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-9)) {
		tmp = U + (J * t_0);
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e-9):
		tmp = U + (J * t_0)
	else:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e-9))
		tmp = Float64(U + Float64(J * t_0));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e-9)))
		tmp = U + (J * t_0);
	else
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-9]], $MachinePrecision]], N[(U + N[(J * t$95$0), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 1.00000000000000006e-9 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 74.6%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.00000000000000006e-9

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 10^{-9}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_2 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -0.001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(J, t\_0 \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* J (* 0.3333333333333333 (pow l 3.0))))))
        (t_2 (+ U (* J (- (exp l) (exp (- l)))))))
   (if (<= l -2.8e+74)
     t_1
     (if (<= l -0.001)
       t_2
       (if (<= l 8e-7)
         (fma J (* t_0 (* 2.0 l)) U)
         (if (<= l 1.85e+90) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (J * (0.3333333333333333 * pow(l, 3.0))));
	double t_2 = U + (J * (exp(l) - exp(-l)));
	double tmp;
	if (l <= -2.8e+74) {
		tmp = t_1;
	} else if (l <= -0.001) {
		tmp = t_2;
	} else if (l <= 8e-7) {
		tmp = fma(J, (t_0 * (2.0 * l)), U);
	} else if (l <= 1.85e+90) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))))
	t_2 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (l <= -2.8e+74)
		tmp = t_1;
	elseif (l <= -0.001)
		tmp = t_2;
	elseif (l <= 8e-7)
		tmp = fma(J, Float64(t_0 * Float64(2.0 * l)), U);
	elseif (l <= 1.85e+90)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return 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[(U + N[(t$95$0 * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.8e+74], t$95$1, If[LessEqual[l, -0.001], t$95$2, If[LessEqual[l, 8e-7], N[(J * N[(t$95$0 * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.85e+90], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.80000000000000002e74 or 1.85e90 < 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 95.9%

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

   4. Taylor expanded in l around inf 95.9%

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

      1. associate-*r* 95.9%

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

      2. *-commutative 95.9%

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

      3. associate-*r* 95.9%

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

   6. Simplified 95.9%

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

   if -2.80000000000000002e74 < l < -1e-3 or 7.9999999999999996e-7 < l < 1.85e90

   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 K around 0 71.9%

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

   if -1e-3 < l < 7.9999999999999996e-7

   1. Initial program 75.2%

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

      1. associate-*l* 75.2%

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

      2. fma-define 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in l around 0 99.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+74}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.001:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \ell\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.06:\\ \;\;\;\;U \cdot \left(1 + \frac{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.06)
   (* U (+ 1.0 (/ (* (* l (cos (* 0.5 K))) (* 2.0 J)) U)))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.06) {
		tmp = U * (1.0 + (((l * cos((0.5 * K))) * (2.0 * J)) / U));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(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)) <= (-0.06d0)) then
        tmp = u * (1.0d0 + (((l * cos((0.5d0 * k))) * (2.0d0 * j)) / u))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (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)) <= -0.06) {
		tmp = U * (1.0 + (((l * Math.cos((0.5 * K))) * (2.0 * J)) / U));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.06:
		tmp = U * (1.0 + (((l * math.cos((0.5 * K))) * (2.0 * J)) / U))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.06)
		tmp = Float64(U * Float64(1.0 + Float64(Float64(Float64(l * cos(Float64(0.5 * K))) * Float64(2.0 * J)) / U)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.06)
		tmp = U * (1.0 + (((l * cos((0.5 * K))) * (2.0 * J)) / U));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.06], N[(U * N[(1.0 + N[(N[(N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.059999999999999998

   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 65.5%

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

   4. Taylor expanded in U around inf 70.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.0%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}{U}}\right) \]

      2. associate-*r* 70.0%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}}{U}\right) \]

      3. *-commutative 70.0%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot 2\right)} \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right) \]

      4. *-commutative 70.0%

         \[\leadsto U \cdot \left(1 + \frac{\left(J \cdot 2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)}}{U}\right) \]

   6. Simplified 70.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{\left(J \cdot 2\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)}{U}\right)} \]

   if -0.059999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.5%

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

   3. Taylor expanded in l around 0 90.1%

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

   4. Taylor expanded in K around 0 86.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.06:\\ \;\;\;\;U \cdot \left(1 + \frac{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot J\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.000115 \lor \neg \left(\ell \leq 8 \cdot 10^{-7}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \ell\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.000115) (not (<= l 8e-7)))
   (+ U (* J (- (exp l) (exp (- l)))))
   (fma J (* (cos (/ K 2.0)) (* 2.0 l)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.000115) || !(l <= 8e-7)) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = fma(J, (cos((K / 2.0)) * (2.0 * l)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.000115) || !(l <= 8e-7))
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = fma(J, Float64(cos(Float64(K / 2.0)) * Float64(2.0 * l)), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.000115], N[Not[LessEqual[l, 8e-7]], $MachinePrecision]], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.15e-4 or 7.9999999999999996e-7 < 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 K around 0 74.6%

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

   if -1.15e-4 < l < 7.9999999999999996e-7

   1. Initial program 75.2%

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

      1. associate-*l* 75.2%

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

      2. fma-define 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in l around 0 99.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.000115 \lor \neg \left(\ell \leq 8 \cdot 10^{-7}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \ell\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.06:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.06)
   (+ U (* l (* (cos (* 0.5 K)) (* 2.0 J))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.06) {
		tmp = U + (l * (cos((0.5 * K)) * (2.0 * J)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(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)) <= (-0.06d0)) then
        tmp = u + (l * (cos((0.5d0 * k)) * (2.0d0 * j)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (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)) <= -0.06) {
		tmp = U + (l * (Math.cos((0.5 * K)) * (2.0 * J)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.06:
		tmp = U + (l * (math.cos((0.5 * K)) * (2.0 * J)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.06)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(0.5 * K)) * Float64(2.0 * J))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.06)
		tmp = U + (l * (cos((0.5 * K)) * (2.0 * J)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.06], N[(U + N[(l * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.059999999999999998

   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 83.0%

      \[\leadsto \color{blue}{\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 83.0%

         \[\leadsto \ell \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      2. associate-*r* 83.0%

         \[\leadsto \ell \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      3. associate-*r* 83.0%

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\left(0.3333333333333333 \cdot {\ell}^{2}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \cdot J + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      4. *-commutative 83.0%

         \[\leadsto \ell \cdot \left(\left(\color{blue}{\left({\ell}^{2} \cdot 0.3333333333333333\right)} \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      5. associate-*r* 83.0%

         \[\leadsto \ell \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \cdot J + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      6. *-commutative 83.0%

         \[\leadsto \ell \cdot \left(\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J + 2 \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right)}\right) + U \]

      7. associate-*r* 83.0%

         \[\leadsto \ell \cdot \left(\left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J + \color{blue}{\left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J}\right) + U \]

      8. distribute-rgt-out 83.0%

         \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left({\ell}^{2} \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right) + 2 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

      9. associate-*r* 83.0%

         \[\leadsto \ell \cdot \left(J \cdot \left(\color{blue}{\left({\ell}^{2} \cdot 0.3333333333333333\right) \cdot \cos \left(0.5 \cdot K\right)} + 2 \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      10. *-commutative 83.0%

          \[\leadsto \ell \cdot \left(J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2}\right)} \cdot \cos \left(0.5 \cdot K\right) + 2 \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      11. distribute-rgt-out 83.0%

          \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{2} + 2\right)\right)}\right) + U \]

      12. +-commutative 83.0%

          \[\leadsto \ell \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)}\right)\right) + U \]

      13. *-commutative 83.0%

          \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(\left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right) \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

   5. Simplified 83.0%

      \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   6. Taylor expanded in l around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*l* 65.5%

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

   8. Simplified 65.5%

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

   if -0.059999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.5%

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

   3. Taylor expanded in l around 0 90.1%

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

   4. Taylor expanded in K around 0 86.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.06:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) * cos((k / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in l around 0 88.8%

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

4. Final simplification 88.8%

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

Alternative 8: 54.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq -4200000:\\ \;\;\;\;\frac{0.25 + \frac{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}{U}}{U}\\ \mathbf{elif}\;\ell \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{-U}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.12e+104)
   (+ U (* J (* 2.0 l)))
   (if (<= l -4200000.0)
     (/ (+ 0.25 (/ (+ (/ (+ 0.140625 (/ 0.10546875 U)) U) -0.1875) U)) U)
     (if (<= l 700.0)
       (fma l (* 2.0 J) U)
       (/ (+ 0.25 (/ (+ 0.1875 (/ -0.140625 U)) U)) (- U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.12e+104) {
		tmp = U + (J * (2.0 * l));
	} else if (l <= -4200000.0) {
		tmp = (0.25 + ((((0.140625 + (0.10546875 / U)) / U) + -0.1875) / U)) / U;
	} else if (l <= 700.0) {
		tmp = fma(l, (2.0 * J), U);
	} else {
		tmp = (0.25 + ((0.1875 + (-0.140625 / U)) / U)) / -U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.12e+104)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	elseif (l <= -4200000.0)
		tmp = Float64(Float64(0.25 + Float64(Float64(Float64(Float64(0.140625 + Float64(0.10546875 / U)) / U) + -0.1875) / U)) / U);
	elseif (l <= 700.0)
		tmp = fma(l, Float64(2.0 * J), U);
	else
		tmp = Float64(Float64(0.25 + Float64(Float64(0.1875 + Float64(-0.140625 / U)) / U)) / Float64(-U));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.12e+104], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4200000.0], N[(N[(0.25 + N[(N[(N[(N[(0.140625 + N[(0.10546875 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] + -0.1875), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 700.0], N[(l * N[(2.0 * J), $MachinePrecision] + U), $MachinePrecision], N[(N[(0.25 + N[(N[(0.1875 + N[(-0.140625 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] / (-U)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.12000000000000003e104

   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 35.6%

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

   4. Taylor expanded in K around 0 25.7%

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

      1. associate-*r* 25.7%

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

      2. *-commutative 25.7%

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

      3. associate-*l* 25.7%

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

   6. Simplified 25.7%

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

   if -1.12000000000000003e104 < l < -4.2e6

   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 1.4%

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

      1. associate-+r+ 1.4%

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

      2. distribute-rgt1-in 1.4%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.4%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.4%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.4%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.4%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.4%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.4%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around -inf 50.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.25 + -1 \cdot \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}{U}} \]
   8. Step-by-step derivation

      1. mul-1-neg 50.5%

         \[\leadsto \color{blue}{-\frac{0.25 + -1 \cdot \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}{U}} \]

      2. distribute-neg-frac2 50.5%

         \[\leadsto \color{blue}{\frac{0.25 + -1 \cdot \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}{-U}} \]

      3. mul-1-neg 50.5%

         \[\leadsto \frac{0.25 + \color{blue}{\left(-\frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}\right)}}{-U} \]

      4. unsub-neg 50.5%

         \[\leadsto \frac{\color{blue}{0.25 - \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}}{-U} \]

      5. mul-1-neg 50.5%

         \[\leadsto \frac{0.25 - \frac{0.1875 + \color{blue}{\left(-\frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}\right)}}{U}}{-U} \]

      6. unsub-neg 50.5%

         \[\leadsto \frac{0.25 - \frac{\color{blue}{0.1875 - \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}}{U}}{-U} \]

      7. associate-*r/ 50.5%

         \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \color{blue}{\frac{0.10546875 \cdot 1}{U}}}{U}}{U}}{-U} \]

      8. metadata-eval 50.5%

         \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{\color{blue}{0.10546875}}{U}}{U}}{U}}{-U} \]

   9. Simplified 50.5%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{-U}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 36.7%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{\sqrt{-U} \cdot \sqrt{-U}}} \]

       2. sqrt-unprod 37.2%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{\sqrt{\left(-U\right) \cdot \left(-U\right)}}} \]

       3. sqr-neg 37.2%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\sqrt{\color{blue}{U \cdot U}}} \]

       4. sqrt-unprod 0.5%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{\sqrt{U} \cdot \sqrt{U}}} \]

       5. add-sqr-sqrt 14.6%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{U}} \]

       6. div-inv 14.6%

          \[\leadsto \color{blue}{\left(0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}\right) \cdot \frac{1}{U}} \]

   11. Applied egg-rr 50.6%

       \[\leadsto \color{blue}{\left(0.25 + \frac{-0.1875 + \frac{0.140625 + \frac{0.10546875}{U}}{U}}{U}\right) \cdot \frac{1}{U}} \]
   12. Step-by-step derivation

       1. associate-*r/ 50.6%

          \[\leadsto \color{blue}{\frac{\left(0.25 + \frac{-0.1875 + \frac{0.140625 + \frac{0.10546875}{U}}{U}}{U}\right) \cdot 1}{U}} \]

       2. *-rgt-identity 50.6%

          \[\leadsto \frac{\color{blue}{0.25 + \frac{-0.1875 + \frac{0.140625 + \frac{0.10546875}{U}}{U}}{U}}}{U} \]

       3. +-commutative 50.6%

          \[\leadsto \frac{0.25 + \frac{\color{blue}{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}}{U}}{U} \]

   13. Simplified 50.6%

       \[\leadsto \color{blue}{\frac{0.25 + \frac{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}{U}}{U}} \]

   if -4.2e6 < l < 700

   1. Initial program 75.6%

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

   3. Taylor expanded in l around 0 99.2%

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

   4. Taylor expanded in K around 0 87.6%

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

      1. +-commutative 87.6%

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

      2. associate-*r* 87.6%

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

      3. *-commutative 87.6%

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

      4. fma-define 87.6%

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

   6. Simplified 87.6%

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

   if 700 < 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

   4. Applied egg-rr 1.9%

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

      1. associate-+r+ 1.9%

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

      2. distribute-rgt1-in 1.9%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.9%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.9%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.9%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.9%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.9%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.9%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.9%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.9%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around -inf 8.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.25 + -1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}{U}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 8.2%

         \[\leadsto -1 \cdot \color{blue}{\left(1 \cdot \frac{0.25 + -1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}{U}\right)} \]

      2. add-sqr-sqrt 7.9%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\sqrt{-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} \cdot \sqrt{-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}}{U}\right) \]

      3. sqrt-unprod 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\sqrt{\left(-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right) \cdot \left(-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)}}}{U}\right) \]

      4. mul-1-neg 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \sqrt{\color{blue}{\left(-\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)} \cdot \left(-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)}}{U}\right) \]

      5. mul-1-neg 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \sqrt{\left(-\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right) \cdot \color{blue}{\left(-\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)}}}{U}\right) \]

      6. sqr-neg 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \sqrt{\color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U} \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}}{U}\right) \]

      7. sqrt-unprod 0.3%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\sqrt{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} \cdot \sqrt{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}}{U}\right) \]

      8. add-sqr-sqrt 28.1%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}{U}\right) \]

      9. cancel-sign-sub-inv 28.1%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \frac{\color{blue}{0.1875 + \left(-0.140625\right) \cdot \frac{1}{U}}}{U}}{U}\right) \]

      10. un-div-inv 28.1%

          \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \frac{0.1875 + \color{blue}{\frac{-0.140625}{U}}}{U}}{U}\right) \]

      11. metadata-eval 28.1%

          \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \frac{0.1875 + \frac{\color{blue}{-0.140625}}{U}}{U}}{U}\right) \]

   9. Applied egg-rr 28.1%

      \[\leadsto -1 \cdot \color{blue}{\left(1 \cdot \frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{U}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 28.1%

          \[\leadsto -1 \cdot \color{blue}{\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{U}} \]

   11. Simplified 28.1%

       \[\leadsto -1 \cdot \color{blue}{\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{U}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq -4200000:\\ \;\;\;\;\frac{0.25 + \frac{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}{U}}{U}\\ \mathbf{elif}\;\ell \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\ell, 2 \cdot J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{-U}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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((0.5d0 * k)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in l around 0 65.9%

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

4. Final simplification 65.9%

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

Alternative 10: 54.4% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -9500000:\\ \;\;\;\;\frac{0.25 + \frac{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}{U}}{U}\\ \mathbf{elif}\;\ell \leq 520:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{-U}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* 2.0 l)))))
   (if (<= l -3.8e+101)
     t_0
     (if (<= l -9500000.0)
       (/ (+ 0.25 (/ (+ (/ (+ 0.140625 (/ 0.10546875 U)) U) -0.1875) U)) U)
       (if (<= l 520.0)
         t_0
         (/ (+ 0.25 (/ (+ 0.1875 (/ -0.140625 U)) U)) (- U)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (2.0 * l));
	double tmp;
	if (l <= -3.8e+101) {
		tmp = t_0;
	} else if (l <= -9500000.0) {
		tmp = (0.25 + ((((0.140625 + (0.10546875 / U)) / U) + -0.1875) / U)) / U;
	} else if (l <= 520.0) {
		tmp = t_0;
	} else {
		tmp = (0.25 + ((0.1875 + (-0.140625 / U)) / U)) / -U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (2.0d0 * l))
    if (l <= (-3.8d+101)) then
        tmp = t_0
    else if (l <= (-9500000.0d0)) then
        tmp = (0.25d0 + ((((0.140625d0 + (0.10546875d0 / u)) / u) + (-0.1875d0)) / u)) / u
    else if (l <= 520.0d0) then
        tmp = t_0
    else
        tmp = (0.25d0 + ((0.1875d0 + ((-0.140625d0) / u)) / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (2.0 * l));
	double tmp;
	if (l <= -3.8e+101) {
		tmp = t_0;
	} else if (l <= -9500000.0) {
		tmp = (0.25 + ((((0.140625 + (0.10546875 / U)) / U) + -0.1875) / U)) / U;
	} else if (l <= 520.0) {
		tmp = t_0;
	} else {
		tmp = (0.25 + ((0.1875 + (-0.140625 / U)) / U)) / -U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (2.0 * l))
	tmp = 0
	if l <= -3.8e+101:
		tmp = t_0
	elif l <= -9500000.0:
		tmp = (0.25 + ((((0.140625 + (0.10546875 / U)) / U) + -0.1875) / U)) / U
	elif l <= 520.0:
		tmp = t_0
	else:
		tmp = (0.25 + ((0.1875 + (-0.140625 / U)) / U)) / -U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(2.0 * l)))
	tmp = 0.0
	if (l <= -3.8e+101)
		tmp = t_0;
	elseif (l <= -9500000.0)
		tmp = Float64(Float64(0.25 + Float64(Float64(Float64(Float64(0.140625 + Float64(0.10546875 / U)) / U) + -0.1875) / U)) / U);
	elseif (l <= 520.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.25 + Float64(Float64(0.1875 + Float64(-0.140625 / U)) / U)) / Float64(-U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (2.0 * l));
	tmp = 0.0;
	if (l <= -3.8e+101)
		tmp = t_0;
	elseif (l <= -9500000.0)
		tmp = (0.25 + ((((0.140625 + (0.10546875 / U)) / U) + -0.1875) / U)) / U;
	elseif (l <= 520.0)
		tmp = t_0;
	else
		tmp = (0.25 + ((0.1875 + (-0.140625 / U)) / U)) / -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.8e+101], t$95$0, If[LessEqual[l, -9500000.0], N[(N[(0.25 + N[(N[(N[(N[(0.140625 + N[(0.10546875 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] + -0.1875), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 520.0], t$95$0, N[(N[(0.25 + N[(N[(0.1875 + N[(-0.140625 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] / (-U)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.7999999999999998e101 or -9.5e6 < l < 520

   1. Initial program 81.4%

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

   3. Taylor expanded in l around 0 84.2%

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

   4. Taylor expanded in K around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-*l* 73.0%

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

   6. Simplified 73.0%

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

   if -3.7999999999999998e101 < l < -9.5e6

   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 1.4%

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

      1. associate-+r+ 1.4%

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

      2. distribute-rgt1-in 1.4%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.4%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.4%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.4%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.4%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.4%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.4%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around -inf 50.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.25 + -1 \cdot \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}{U}} \]
   8. Step-by-step derivation

      1. mul-1-neg 50.5%

         \[\leadsto \color{blue}{-\frac{0.25 + -1 \cdot \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}{U}} \]

      2. distribute-neg-frac2 50.5%

         \[\leadsto \color{blue}{\frac{0.25 + -1 \cdot \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}{-U}} \]

      3. mul-1-neg 50.5%

         \[\leadsto \frac{0.25 + \color{blue}{\left(-\frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}\right)}}{-U} \]

      4. unsub-neg 50.5%

         \[\leadsto \frac{\color{blue}{0.25 - \frac{0.1875 + -1 \cdot \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}{U}}}{-U} \]

      5. mul-1-neg 50.5%

         \[\leadsto \frac{0.25 - \frac{0.1875 + \color{blue}{\left(-\frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}\right)}}{U}}{-U} \]

      6. unsub-neg 50.5%

         \[\leadsto \frac{0.25 - \frac{\color{blue}{0.1875 - \frac{0.140625 - 0.10546875 \cdot \frac{1}{U}}{U}}}{U}}{-U} \]

      7. associate-*r/ 50.5%

         \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \color{blue}{\frac{0.10546875 \cdot 1}{U}}}{U}}{U}}{-U} \]

      8. metadata-eval 50.5%

         \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{\color{blue}{0.10546875}}{U}}{U}}{U}}{-U} \]

   9. Simplified 50.5%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{-U}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 36.7%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{\sqrt{-U} \cdot \sqrt{-U}}} \]

       2. sqrt-unprod 37.2%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{\sqrt{\left(-U\right) \cdot \left(-U\right)}}} \]

       3. sqr-neg 37.2%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\sqrt{\color{blue}{U \cdot U}}} \]

       4. sqrt-unprod 0.5%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{\sqrt{U} \cdot \sqrt{U}}} \]

       5. add-sqr-sqrt 14.6%

          \[\leadsto \frac{0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}}{\color{blue}{U}} \]

       6. div-inv 14.6%

          \[\leadsto \color{blue}{\left(0.25 - \frac{0.1875 - \frac{0.140625 - \frac{0.10546875}{U}}{U}}{U}\right) \cdot \frac{1}{U}} \]

   11. Applied egg-rr 50.6%

       \[\leadsto \color{blue}{\left(0.25 + \frac{-0.1875 + \frac{0.140625 + \frac{0.10546875}{U}}{U}}{U}\right) \cdot \frac{1}{U}} \]
   12. Step-by-step derivation

       1. associate-*r/ 50.6%

          \[\leadsto \color{blue}{\frac{\left(0.25 + \frac{-0.1875 + \frac{0.140625 + \frac{0.10546875}{U}}{U}}{U}\right) \cdot 1}{U}} \]

       2. *-rgt-identity 50.6%

          \[\leadsto \frac{\color{blue}{0.25 + \frac{-0.1875 + \frac{0.140625 + \frac{0.10546875}{U}}{U}}{U}}}{U} \]

       3. +-commutative 50.6%

          \[\leadsto \frac{0.25 + \frac{\color{blue}{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}}{U}}{U} \]

   13. Simplified 50.6%

       \[\leadsto \color{blue}{\frac{0.25 + \frac{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}{U}}{U}} \]

   if 520 < 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

   4. Applied egg-rr 1.9%

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

      1. associate-+r+ 1.9%

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

      2. distribute-rgt1-in 1.9%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.9%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.9%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.9%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.9%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.9%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.9%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.9%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.9%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around -inf 8.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.25 + -1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}{U}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 8.2%

         \[\leadsto -1 \cdot \color{blue}{\left(1 \cdot \frac{0.25 + -1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}{U}\right)} \]

      2. add-sqr-sqrt 7.9%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\sqrt{-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} \cdot \sqrt{-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}}{U}\right) \]

      3. sqrt-unprod 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\sqrt{\left(-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right) \cdot \left(-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)}}}{U}\right) \]

      4. mul-1-neg 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \sqrt{\color{blue}{\left(-\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)} \cdot \left(-1 \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)}}{U}\right) \]

      5. mul-1-neg 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \sqrt{\left(-\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right) \cdot \color{blue}{\left(-\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}\right)}}}{U}\right) \]

      6. sqr-neg 8.2%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \sqrt{\color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U} \cdot \frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}}{U}\right) \]

      7. sqrt-unprod 0.3%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\sqrt{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} \cdot \sqrt{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}}{U}\right) \]

      8. add-sqr-sqrt 28.1%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}}}{U}\right) \]

      9. cancel-sign-sub-inv 28.1%

         \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \frac{\color{blue}{0.1875 + \left(-0.140625\right) \cdot \frac{1}{U}}}{U}}{U}\right) \]

      10. un-div-inv 28.1%

          \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \frac{0.1875 + \color{blue}{\frac{-0.140625}{U}}}{U}}{U}\right) \]

      11. metadata-eval 28.1%

          \[\leadsto -1 \cdot \left(1 \cdot \frac{0.25 + \frac{0.1875 + \frac{\color{blue}{-0.140625}}{U}}{U}}{U}\right) \]

   9. Applied egg-rr 28.1%

      \[\leadsto -1 \cdot \color{blue}{\left(1 \cdot \frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{U}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 28.1%

          \[\leadsto -1 \cdot \color{blue}{\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{U}} \]

   11. Simplified 28.1%

       \[\leadsto -1 \cdot \color{blue}{\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{U}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+101}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq -9500000:\\ \;\;\;\;\frac{0.25 + \frac{\frac{0.140625 + \frac{0.10546875}{U}}{U} + -0.1875}{U}}{U}\\ \mathbf{elif}\;\ell \leq 520:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{0.1875 + \frac{-0.140625}{U}}{U}}{-U}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.5% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+23} \lor \neg \left(\ell \leq 1220\right):\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.7e+23) (not (<= l 1220.0))) (* U (- 2.0 U)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.7e+23) || !(l <= 1220.0)) {
		tmp = U * (2.0 - 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 <= (-5.7d+23)) .or. (.not. (l <= 1220.0d0))) then
        tmp = u * (2.0d0 - 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 <= -5.7e+23) || !(l <= 1220.0)) {
		tmp = U * (2.0 - U);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.7e+23) or not (l <= 1220.0):
		tmp = U * (2.0 - U)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.7e+23) || !(l <= 1220.0))
		tmp = Float64(U * Float64(2.0 - U));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -5.7e+23) || ~((l <= 1220.0)))
		tmp = U * (2.0 - U);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.7e+23], N[Not[LessEqual[l, 1220.0]], $MachinePrecision]], N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.7e23 or 1220 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.2%

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

      1. +-commutative 15.2%

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

      2. fma-undefine 15.2%

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

      3. associate-+l+ 15.2%

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

      4. count-2 15.2%

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

      5. distribute-rgt-out 15.2%

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

   8. Simplified 15.2%

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

   9. Taylor expanded in U around 0 15.2%

      \[\leadsto \color{blue}{U \cdot \left(2 + -1 \cdot U\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 15.2%

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

       2. unsub-neg 15.2%

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

   11. Simplified 15.2%

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

   if -5.7e23 < l < 1220

   1. Initial program 76.0%

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

   3. Taylor expanded in J around 0 73.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+23} \lor \neg \left(\ell \leq 1220\right):\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+24} \lor \neg \left(\ell \leq 580\right):\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.05e+24) (not (<= l 580.0))) (- -4.0 (* U U)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.05e+24) || !(l <= 580.0)) {
		tmp = -4.0 - (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 <= (-1.05d+24)) .or. (.not. (l <= 580.0d0))) then
        tmp = (-4.0d0) - (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 <= -1.05e+24) || !(l <= 580.0)) {
		tmp = -4.0 - (U * U);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.05e+24) or not (l <= 580.0):
		tmp = -4.0 - (U * U)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.05e+24) || !(l <= 580.0))
		tmp = Float64(-4.0 - Float64(U * U));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.05e+24) || ~((l <= 580.0)))
		tmp = -4.0 - (U * U);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.05e+24], N[Not[LessEqual[l, 580.0]], $MachinePrecision]], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.0500000000000001e24 or 580 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 15.5%

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

      1. cancel-sign-sub-inv 15.5%

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

   8. Simplified 15.5%

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

   if -1.0500000000000001e24 < l < 580

   1. Initial program 76.1%

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

   3. Taylor expanded in J around 0 72.8%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+24} \lor \neg \left(\ell \leq 580\right):\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.5% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+84}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -3.3e+84) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.3e+84) {
		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 <= (-3.3d+84)) 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 <= -3.3e+84) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.3e+84:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.3e+84)
		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 <= -3.3e+84)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.3e+84], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -3.30000000000000017e84

   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 12.3%

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

   if -3.30000000000000017e84 < l

   1. Initial program 84.3%

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

   3. Taylor expanded in J around 0 48.7%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+84}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.8% accurate, 44.6× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * 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 + (j * (2.0d0 * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in l around 0 65.9%

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

4. Taylor expanded in K around 0 56.3%

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

   1. associate-*r* 56.3%

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

   2. *-commutative 56.3%

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

   3. associate-*l* 56.3%

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

6. Simplified 56.3%

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

7. Final simplification 56.3%

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

Alternative 15: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -0.3333333333333333)

C

double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = -0.3333333333333333d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

Python

def code(J, l, K, U):
	return -0.3333333333333333

Julia

function code(J, l, K, U)
	return -0.3333333333333333
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -0.3333333333333333;
end

Wolfram

code[J_, l_, K_, U_] := -0.3333333333333333

Derivation

1. Initial program 87.0%

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

4. Applied egg-rr 2.4%

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

   1. associate-+r+ 2.4%

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

   2. distribute-rgt1-in 2.4%

      \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

   3. metadata-eval 2.4%

      \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

   4. *-commutative 2.4%

      \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

   5. distribute-lft-out 2.4%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

   6. associate-/r* 2.3%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

   7. *-inverses 2.3%

      \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

   8. +-commutative 2.3%

      \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

   9. *-commutative 2.3%

      \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

6. Simplified 2.3%

   \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

7. Taylor expanded in U around 0 2.9%

   \[\leadsto \color{blue}{-0.3333333333333333} \]

8. Final simplification 2.9%

   \[\leadsto -0.3333333333333333 \]
9. Add Preprocessing

Alternative 16: 37.4% 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 87.0%

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

3. Taylor expanded in J around 0 40.5%

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

4. Final simplification 40.5%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024061 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))