Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 99.7%

Time: 10.2s

Alternatives: 15

Speedup: 2.7×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -20000 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, t\_0 \cdot \left(\ell \cdot 2\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -20000.0) (not (<= t_1 5e-11)))
     (+ (* (* t_1 J) t_0) U)
     (fma J (* t_0 (* l 2.0)) U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -20000.0) || !(t_1 <= 5e-11)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = fma(J, (t_0 * (l * 2.0)), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -20000.0) || !(t_1 <= 5e-11))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = fma(J, Float64(t_0 * Float64(l * 2.0)), U);
	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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -20000.0], N[Not[LessEqual[t$95$1, 5e-11]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(t$95$0 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -2e4 or 5.00000000000000018e-11 < (-.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

   if -2e4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.00000000000000018e-11

   1. Initial program 66.6%

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

      1. associate-*l* 66.6%

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

      2. fma-define 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -20000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.53:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.0041:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.53)
     (+ U (* J (* l (+ 2.0 (* (pow K 2.0) -0.25)))))
     (if (<= t_0 0.0041)
       (+ U (* t_0 (* l (* J 2.0))))
       (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.53) {
		tmp = U + (J * (l * (2.0 + (pow(K, 2.0) * -0.25))));
	} else if (t_0 <= 0.0041) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.53d0)) then
        tmp = u + (j * (l * (2.0d0 + ((k ** 2.0d0) * (-0.25d0)))))
    else if (t_0 <= 0.0041d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    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 t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.53) {
		tmp = U + (J * (l * (2.0 + (Math.pow(K, 2.0) * -0.25))));
	} else if (t_0 <= 0.0041) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.53:
		tmp = U + (J * (l * (2.0 + (math.pow(K, 2.0) * -0.25))))
	elif t_0 <= 0.0041:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.53)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((K ^ 2.0) * -0.25)))));
	elseif (t_0 <= 0.0041)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	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)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.53)
		tmp = U + (J * (l * (2.0 + ((K ^ 2.0) * -0.25))));
	elseif (t_0 <= 0.0041)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.53], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0041], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $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 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.53000000000000003

   1. Initial program 86.1%

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

   3. Taylor expanded in l around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*l* 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-*l* 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in K around 0 74.2%

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

      1. associate-*r* 74.2%

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

      2. distribute-rgt-out 74.2%

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

      3. *-commutative 74.2%

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

   8. Simplified 74.2%

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

   if -0.53000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.00410000000000000035

   1. Initial program 77.1%

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

   3. Taylor expanded in l around 0 67.7%

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

      1. associate-*r* 67.7%

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

      2. *-commutative 67.7%

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

   5. Simplified 67.7%

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

   if 0.00410000000000000035 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0 87.6%

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

   4. Taylor expanded in K around 0 82.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.53:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.0041:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+87}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -6.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 7:\\ \;\;\;\;\mathsf{fma}\left(J, t\_1 \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U))
        (t_1 (cos (/ K 2.0)))
        (t_2 (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))
   (if (<= l -9e+87)
     t_2
     (if (<= l -6.6)
       t_0
       (if (<= l 7.0)
         (fma J (* t_1 (* l 2.0)) U)
         (if (<= l 2.7e+100) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K / 2.0));
	double t_2 = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double tmp;
	if (l <= -9e+87) {
		tmp = t_2;
	} else if (l <= -6.6) {
		tmp = t_0;
	} else if (l <= 7.0) {
		tmp = fma(J, (t_1 * (l * 2.0)), U);
	} else if (l <= 2.7e+100) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
	tmp = 0.0
	if (l <= -9e+87)
		tmp = t_2;
	elseif (l <= -6.6)
		tmp = t_0;
	elseif (l <= 7.0)
		tmp = fma(J, Float64(t_1 * Float64(l * 2.0)), U);
	elseif (l <= 2.7e+100)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9e+87], t$95$2, If[LessEqual[l, -6.6], t$95$0, If[LessEqual[l, 7.0], N[(J * N[(t$95$1 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 2.7e+100], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.0000000000000005e87 or 2.69999999999999998e100 < 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 99.0%

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

      1. unpow2 99.0%

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

   5. Applied egg-rr 99.0%

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

   if -9.0000000000000005e87 < l < -6.5999999999999996 or 7 < l < 2.69999999999999998e100

   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 77.7%

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

   if -6.5999999999999996 < l < 7

   1. Initial program 67.2%

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

      1. associate-*l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in l around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+87}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -6.6:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 7:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -6.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.55:\\ \;\;\;\;U + J \cdot \left(\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -8.5e+87)
     t_0
     (if (<= l -6.6)
       t_1
       (if (<= l 0.55)
         (+ U (* J (* (* l 2.0) (cos (* K 0.5)))))
         (if (<= l 1.55e+100) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -8.5e+87) {
		tmp = t_0;
	} else if (l <= -6.6) {
		tmp = t_1;
	} else if (l <= 0.55) {
		tmp = U + (J * ((l * 2.0) * cos((K * 0.5))));
	} else if (l <= 1.55e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-8.5d+87)) then
        tmp = t_0
    else if (l <= (-6.6d0)) then
        tmp = t_1
    else if (l <= 0.55d0) then
        tmp = u + (j * ((l * 2.0d0) * cos((k * 0.5d0))))
    else if (l <= 1.55d+100) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -8.5e+87) {
		tmp = t_0;
	} else if (l <= -6.6) {
		tmp = t_1;
	} else if (l <= 0.55) {
		tmp = U + (J * ((l * 2.0) * Math.cos((K * 0.5))));
	} else if (l <= 1.55e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -8.5e+87:
		tmp = t_0
	elif l <= -6.6:
		tmp = t_1
	elif l <= 0.55:
		tmp = U + (J * ((l * 2.0) * math.cos((K * 0.5))))
	elif l <= 1.55e+100:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -8.5e+87)
		tmp = t_0;
	elseif (l <= -6.6)
		tmp = t_1;
	elseif (l <= 0.55)
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) * cos(Float64(K * 0.5)))));
	elseif (l <= 1.55e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -8.5e+87)
		tmp = t_0;
	elseif (l <= -6.6)
		tmp = t_1;
	elseif (l <= 0.55)
		tmp = U + (J * ((l * 2.0) * cos((K * 0.5))));
	elseif (l <= 1.55e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -8.5e+87], t$95$0, If[LessEqual[l, -6.6], t$95$1, If[LessEqual[l, 0.55], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.5000000000000001e87 or 1.55000000000000003e100 < 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 99.0%

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

      1. unpow2 99.0%

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

   5. Applied egg-rr 99.0%

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

   if -8.5000000000000001e87 < l < -6.5999999999999996 or 0.55000000000000004 < l < 1.55000000000000003e100

   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 77.7%

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

   if -6.5999999999999996 < l < 0.55000000000000004

   1. Initial program 67.2%

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

   3. Taylor expanded in l around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*l* 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+87}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -6.6:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.55:\\ \;\;\;\;U + J \cdot \left(\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -6.6:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(J, t\_1 \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - t\_0\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -6.6)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 0.6)
       (fma J (* t_1 (* l 2.0)) U)
       (if (<= l 3e+100)
         (+ (* (- (exp l) t_0) J) U)
         (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -6.6) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.6) {
		tmp = fma(J, (t_1 * (l * 2.0)), U);
	} else if (l <= 3e+100) {
		tmp = ((exp(l) - t_0) * J) + U;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -6.6)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 0.6)
		tmp = fma(J, Float64(t_1 * Float64(l * 2.0)), U);
	elseif (l <= 3e+100)
		tmp = Float64(Float64(Float64(exp(l) - t_0) * J) + U);
	else
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -6.6], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.6], N[(J * N[(t$95$1 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 3e+100], N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.5999999999999996

   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 99.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -6.5999999999999996 < l < 0.599999999999999978

   1. Initial program 67.2%

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

      1. associate-*l* 67.2%

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

      2. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in l around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   if 0.599999999999999978 < l < 2.99999999999999985e100

   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 70.6%

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

   if 2.99999999999999985e100 < 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 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.0041:\\ \;\;\;\;U + J \cdot \left(\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.0041)
   (+ U (* J (* (* l 2.0) (cos (* K 0.5)))))
   (+ 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)) <= 0.0041) {
		tmp = U + (J * ((l * 2.0) * cos((K * 0.5))));
	} 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)) <= 0.0041d0) then
        tmp = u + (j * ((l * 2.0d0) * cos((k * 0.5d0))))
    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)) <= 0.0041) {
		tmp = U + (J * ((l * 2.0) * Math.cos((K * 0.5))));
	} 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)) <= 0.0041:
		tmp = U + (J * ((l * 2.0) * math.cos((K * 0.5))))
	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)) <= 0.0041)
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) * cos(Float64(K * 0.5)))));
	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)) <= 0.0041)
		tmp = U + (J * ((l * 2.0) * cos((K * 0.5))));
	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], 0.0041], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $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))) < 0.00410000000000000035

   1. Initial program 82.7%

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

   3. Taylor expanded in l around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*l* 61.8%

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

   5. Simplified 61.8%

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

   if 0.00410000000000000035 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0 87.6%

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

   4. Taylor expanded in K around 0 82.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.0041:\\ \;\;\;\;U + J \cdot \left(\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * 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 + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in l around 0 89.5%

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

   1. unpow2 89.5%

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

5. Applied egg-rr 89.5%

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

6. Final simplification 89.5%

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

Alternative 8: 64.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (J * ((l * 2.0) * 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 + (j * ((l * 2.0d0) * cos((k * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in l around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-*l* 62.2%

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

   3. *-commutative 62.2%

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

   4. associate-*l* 62.2%

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

5. Simplified 62.2%

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

6. Final simplification 62.2%

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

Alternative 9: 42.9% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+28}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 1020:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-4 - U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.35e+28) (* U (- U -4.0)) (if (<= l 1020.0) U (- -4.0 (* U U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.35e+28) {
		tmp = U * (U - -4.0);
	} else if (l <= 1020.0) {
		tmp = U;
	} else {
		tmp = -4.0 - (U * U);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.35d+28)) then
        tmp = u * (u - (-4.0d0))
    else if (l <= 1020.0d0) then
        tmp = u
    else
        tmp = (-4.0d0) - (u * u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.35e+28) {
		tmp = U * (U - -4.0);
	} else if (l <= 1020.0) {
		tmp = U;
	} else {
		tmp = -4.0 - (U * U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.35e+28:
		tmp = U * (U - -4.0)
	elif l <= 1020.0:
		tmp = U
	else:
		tmp = -4.0 - (U * U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.35e+28)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 1020.0)
		tmp = U;
	else
		tmp = Float64(-4.0 - Float64(U * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.35e+28)
		tmp = U * (U - -4.0);
	elseif (l <= 1020.0)
		tmp = U;
	else
		tmp = -4.0 - (U * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.35e+28], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1020.0], U, N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3500000000000001e28

   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 24.5%

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

   if -1.3500000000000001e28 < l < 1020

   1. Initial program 68.9%

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

      1. associate-*l* 68.9%

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

      2. fma-define 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in J around 0 62.8%

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

   if 1020 < 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 18.4%

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

      1. cancel-sign-sub-inv 18.4%

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

   8. Simplified 18.4%

      \[\leadsto \color{blue}{-4 - U \cdot U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 42.8% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+28}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 0.54:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.35e+28) (* U (- U -4.0)) (if (<= l 0.54) U (* U (- 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.35e+28) {
		tmp = U * (U - -4.0);
	} else if (l <= 0.54) {
		tmp = U;
	} else {
		tmp = U * (2.0 - 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 <= (-1.35d+28)) then
        tmp = u * (u - (-4.0d0))
    else if (l <= 0.54d0) then
        tmp = u
    else
        tmp = u * (2.0d0 - u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.35e+28) {
		tmp = U * (U - -4.0);
	} else if (l <= 0.54) {
		tmp = U;
	} else {
		tmp = U * (2.0 - U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.35e+28:
		tmp = U * (U - -4.0)
	elif l <= 0.54:
		tmp = U
	else:
		tmp = U * (2.0 - U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.35e+28)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 0.54)
		tmp = U;
	else
		tmp = Float64(U * Float64(2.0 - U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.35e+28)
		tmp = U * (U - -4.0);
	elseif (l <= 0.54)
		tmp = U;
	else
		tmp = U * (2.0 - U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.35e+28], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.54], U, N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3500000000000001e28

   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 24.5%

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

   if -1.3500000000000001e28 < l < 0.54000000000000004

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. fma-define 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in J around 0 63.3%

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

   if 0.54000000000000004 < 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 18.0%

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

   7. Taylor expanded in U around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. unsub-neg 18.0%

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

   9. Simplified 18.0%

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

Alternative 11: 42.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+35}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 0.54:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6.6e+35) (* U U) (if (<= l 0.54) U (* U (- 2.0 U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.6e+35) {
		tmp = U * U;
	} else if (l <= 0.54) {
		tmp = U;
	} else {
		tmp = U * (2.0 - 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 <= (-6.6d+35)) then
        tmp = u * u
    else if (l <= 0.54d0) then
        tmp = u
    else
        tmp = u * (2.0d0 - u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.6e+35) {
		tmp = U * U;
	} else if (l <= 0.54) {
		tmp = U;
	} else {
		tmp = U * (2.0 - U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6.6e+35:
		tmp = U * U
	elif l <= 0.54:
		tmp = U
	else:
		tmp = U * (2.0 - U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6.6e+35)
		tmp = Float64(U * U);
	elseif (l <= 0.54)
		tmp = U;
	else
		tmp = Float64(U * Float64(2.0 - U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -6.6e+35)
		tmp = U * U;
	elseif (l <= 0.54)
		tmp = U;
	else
		tmp = U * (2.0 - U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6.6e+35], N[(U * U), $MachinePrecision], If[LessEqual[l, 0.54], U, N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.6000000000000003e35

   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 24.7%

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

   if -6.6000000000000003e35 < l < 0.54000000000000004

   1. Initial program 68.9%

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

      1. associate-*l* 68.9%

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

      2. fma-define 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in J around 0 62.8%

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

   if 0.54000000000000004 < 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 18.0%

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

   7. Taylor expanded in U around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. unsub-neg 18.0%

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

   9. Simplified 18.0%

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

Alternative 12: 42.8% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+35} \lor \neg \left(\ell \leq 15000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.6e+35) (not (<= l 15000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.6e+35) || !(l <= 15000.0)) {
		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 <= (-6.6d+35)) .or. (.not. (l <= 15000.0d0))) 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 <= -6.6e+35) || !(l <= 15000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.6e+35) or not (l <= 15000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.6e+35) || !(l <= 15000.0))
		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 <= -6.6e+35) || ~((l <= 15000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.6e+35], N[Not[LessEqual[l, 15000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.6000000000000003e35 or 15000 < 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 20.5%

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

   if -6.6000000000000003e35 < l < 15000

   1. Initial program 69.4%

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

      1. associate-*l* 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in J around 0 61.9%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+35} \lor \neg \left(\ell \leq 15000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in l around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-*l* 62.2%

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

   3. *-commutative 62.2%

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

   4. associate-*l* 62.2%

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

5. Simplified 62.2%

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

6. Taylor expanded in K around 0 50.5%

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

   1. associate-*r* 50.5%

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

   2. *-commutative 50.5%

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

8. Simplified 50.5%

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

9. Final simplification 50.5%

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

Alternative 14: 37.1% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. associate-*l* 84.2%

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

   2. fma-define 84.2%

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

3. Simplified 84.2%

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

5. Taylor expanded in J around 0 33.0%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

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 = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 84.2%

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

   1. associate-*l* 84.2%

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

   2. fma-define 84.2%

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

3. Simplified 84.2%

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

6. Applied egg-rr 24.0%

   \[\leadsto \color{blue}{-4 + U} \]
7. Step-by-step derivation

   1. +-commutative 24.0%

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

8. Simplified 24.0%

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

9. Taylor expanded in U around 0 2.7%

   \[\leadsto \color{blue}{-4} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))