Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.8%

Time: 22.0s

Alternatives: 18

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_1 := K \cdot \left(m + n\right)\\ \mathbf{if}\;\cos \left(\frac{t\_1}{2} - M\right) \cdot t\_0 \leq 2:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{{\left(\sqrt[3]{t\_1}\right)}^{3}}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{m - \left(\left(n + \ell\right) + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0))))
        (t_1 (* K (+ m n))))
   (if (<= (* (cos (- (/ t_1 2.0) M)) t_0) 2.0)
     (* t_0 (cos (- (/ (pow (cbrt t_1) 3.0) 2.0) M)))
     (exp (- m (+ (+ n l) (* 0.25 (pow (+ m n) 2.0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double t_1 = K * (m + n);
	double tmp;
	if ((cos(((t_1 / 2.0) - M)) * t_0) <= 2.0) {
		tmp = t_0 * cos(((pow(cbrt(t_1), 3.0) / 2.0) - M));
	} else {
		tmp = exp((m - ((n + l) + (0.25 * pow((m + n), 2.0)))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double t_1 = K * (m + n);
	double tmp;
	if ((Math.cos(((t_1 / 2.0) - M)) * t_0) <= 2.0) {
		tmp = t_0 * Math.cos(((Math.pow(Math.cbrt(t_1), 3.0) / 2.0) - M));
	} else {
		tmp = Math.exp((m - ((n + l) + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	t_1 = Float64(K * Float64(m + n))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(t_1 / 2.0) - M)) * t_0) <= 2.0)
		tmp = Float64(t_0 * cos(Float64(Float64((cbrt(t_1) ^ 3.0) / 2.0) - M)));
	else
		tmp = exp(Float64(m - Float64(Float64(n + l) + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(t$95$1 / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(t$95$0 * N[Cos[N[(N[(N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(m - N[(N[(n + l), $MachinePrecision] + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < 2

   1. Initial program 96.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 96.6%

         \[\leadsto \cos \left(\frac{\color{blue}{\left(\sqrt[3]{K \cdot \left(m + n\right)} \cdot \sqrt[3]{K \cdot \left(m + n\right)}\right) \cdot \sqrt[3]{K \cdot \left(m + n\right)}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. pow3 97.1%

         \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   4. Applied egg-rr 97.1%

      \[\leadsto \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if 2 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 16.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg 16.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\left(\ell - \left|m - n\right|\right)\right)}} \]

      2. distribute-neg-out 16.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)}} \]

      3. div-inv 16.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      4. metadata-eval 16.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

      5. add-sqr-sqrt 6.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right|\right)\right)} \]

      6. fabs-sqr 6.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right)\right)} \]

      7. add-sqr-sqrt 16.7%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\left(m - n\right)}\right)\right)} \]

   4. Applied egg-rr 16.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left(m - n\right)\right)\right)}} \]

   5. Taylor expanded in K around 0 96.7%

      \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{m - \left(\ell + \left(n + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)\right)}} \]

   7. Simplified 96.7%

      \[\leadsto \color{blue}{\cos M \cdot e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}} \]

   8. Taylor expanded in M around 0 98.3%

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

      1. associate-+r+ 98.3%

         \[\leadsto e^{m - \color{blue}{\left(\left(\ell + n\right) + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

      2. +-commutative 98.3%

         \[\leadsto e^{m - \left(\left(\ell + n\right) + 0.25 \cdot {\color{blue}{\left(n + m\right)}}^{2}\right)} \]

   10. Simplified 98.3%

       \[\leadsto \color{blue}{e^{m - \left(\left(\ell + n\right) + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq 2:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{{\left(\sqrt[3]{K \cdot \left(m + n\right)}\right)}^{3}}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{m - \left(\left(n + \ell\right) + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- M))
  (exp (- (fabs (- m n)) (+ l (pow (- (* (+ m n) 0.5) M) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(-M) * exp((fabs((m - n)) - (l + pow((((m + n) * 0.5) - M), 2.0))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(-m_1) * exp((abs((m - n)) - (l + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(-M) * Math.exp((Math.abs((m - n)) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

def code(K, m, n, M, l):
	return math.cos(-M) * math.exp((math.fabs((m - n)) - (l + math.pow((((m + n) * 0.5) - M), 2.0))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(-M)) * exp(Float64(abs(Float64(m - n)) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(-M) * exp((abs((m - n)) - (l + ((((m + n) * 0.5) - M) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[(-M)], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in K around 0 95.8%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

4. Final simplification 95.8%

   \[\leadsto \cos \left(-M\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- m l) (+ n (pow (- (* (+ m n) 0.5) M) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp(((m - l) - (n + pow((((m + n) * 0.5) - M), 2.0))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(((m - l) - (n + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp(((m - l) - (n + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp(((m - l) - (n + math.pow((((m + n) * 0.5) - M), 2.0))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(Float64(m - l) - Float64(n + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(((m - l) - (n + ((((m + n) * 0.5) - M) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - l), $MachinePrecision] - N[(n + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\left(\ell - \left|m - n\right|\right)\right)}} \]

   2. distribute-neg-out 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)}} \]

   3. div-inv 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   4. metadata-eval 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   5. add-sqr-sqrt 36.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right|\right)\right)} \]

   6. fabs-sqr 36.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right)\right)} \]

   7. add-sqr-sqrt 77.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\left(m - n\right)}\right)\right)} \]

4. Applied egg-rr 77.6%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left(m - n\right)\right)\right)}} \]

5. Taylor expanded in K around 0 95.7%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{m - \left(\ell + \left(n + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)\right)}} \]

7. Simplified 95.7%

   \[\leadsto \color{blue}{\cos M \cdot e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}} \]
8. Add Preprocessing

Alternative 4: 38.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+96}:\\ \;\;\;\;\cos \left(\frac{K \cdot n}{2} - M\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= l -2.3e+96)
     (* (cos (- (/ (* K n) 2.0) M)) t_0)
     (if (<= l 3.9e-125)
       (* 0.5 (* K (* m (* (sin M) (- 1.0 l)))))
       (* (cos M) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if (l <= -2.3e+96) {
		tmp = cos((((K * n) / 2.0) - M)) * t_0;
	} else if (l <= 3.9e-125) {
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	} else {
		tmp = cos(M) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-l)
    if (l <= (-2.3d+96)) then
        tmp = cos((((k * n) / 2.0d0) - m_1)) * t_0
    else if (l <= 3.9d-125) then
        tmp = 0.5d0 * (k * (m * (sin(m_1) * (1.0d0 - l))))
    else
        tmp = cos(m_1) * t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-l);
	double tmp;
	if (l <= -2.3e+96) {
		tmp = Math.cos((((K * n) / 2.0) - M)) * t_0;
	} else if (l <= 3.9e-125) {
		tmp = 0.5 * (K * (m * (Math.sin(M) * (1.0 - l))));
	} else {
		tmp = Math.cos(M) * t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if l <= -2.3e+96:
		tmp = math.cos((((K * n) / 2.0) - M)) * t_0
	elif l <= 3.9e-125:
		tmp = 0.5 * (K * (m * (math.sin(M) * (1.0 - l))))
	else:
		tmp = math.cos(M) * t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -2.3e+96)
		tmp = Float64(cos(Float64(Float64(Float64(K * n) / 2.0) - M)) * t_0);
	elseif (l <= 3.9e-125)
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * Float64(1.0 - l)))));
	else
		tmp = Float64(cos(M) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if (l <= -2.3e+96)
		tmp = cos((((K * n) / 2.0) - M)) * t_0;
	elseif (l <= 3.9e-125)
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	else
		tmp = cos(M) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -2.3e+96], N[(N[Cos[N[(N[(N[(K * n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 3.9e-125], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.30000000000000015e96

   1. Initial program 79.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 32.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 32.6%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in m around 0 30.3%

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

   if -2.30000000000000015e96 < l < 3.89999999999999982e-125

   1. Initial program 78.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 11.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 11.7%

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

   5. Simplified 11.7%

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

   6. Taylor expanded in l around 0 11.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 11.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 11.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 11.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 10.5%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 10.5%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 10.5%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 10.5%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 10.5%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 18.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]

   if 3.89999999999999982e-125 < l

   1. Initial program 75.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 56.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in K around 0 72.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. cos-neg 72.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

   8. Simplified 72.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 38.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+147}:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{K \cdot m}{2} - M\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= l -2.4e+147)
     (* t_0 (cos (- (/ (* K m) 2.0) M)))
     (if (<= l 5.5e-126)
       (* 0.5 (* K (* m (* (sin M) (- 1.0 l)))))
       (* (cos M) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if (l <= -2.4e+147) {
		tmp = t_0 * cos((((K * m) / 2.0) - M));
	} else if (l <= 5.5e-126) {
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	} else {
		tmp = cos(M) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-l)
    if (l <= (-2.4d+147)) then
        tmp = t_0 * cos((((k * m) / 2.0d0) - m_1))
    else if (l <= 5.5d-126) then
        tmp = 0.5d0 * (k * (m * (sin(m_1) * (1.0d0 - l))))
    else
        tmp = cos(m_1) * t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-l);
	double tmp;
	if (l <= -2.4e+147) {
		tmp = t_0 * Math.cos((((K * m) / 2.0) - M));
	} else if (l <= 5.5e-126) {
		tmp = 0.5 * (K * (m * (Math.sin(M) * (1.0 - l))));
	} else {
		tmp = Math.cos(M) * t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if l <= -2.4e+147:
		tmp = t_0 * math.cos((((K * m) / 2.0) - M))
	elif l <= 5.5e-126:
		tmp = 0.5 * (K * (m * (math.sin(M) * (1.0 - l))))
	else:
		tmp = math.cos(M) * t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -2.4e+147)
		tmp = Float64(t_0 * cos(Float64(Float64(Float64(K * m) / 2.0) - M)));
	elseif (l <= 5.5e-126)
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * Float64(1.0 - l)))));
	else
		tmp = Float64(cos(M) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if (l <= -2.4e+147)
		tmp = t_0 * cos((((K * m) / 2.0) - M));
	elseif (l <= 5.5e-126)
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	else
		tmp = cos(M) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -2.4e+147], N[(t$95$0 * N[Cos[N[(N[(N[(K * m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e-126], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.40000000000000002e147

   1. Initial program 81.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 40.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 40.1%

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

   5. Simplified 40.1%

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

   6. Taylor expanded in m around inf 40.2%

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

   if -2.40000000000000002e147 < l < 5.49999999999999987e-126

   1. Initial program 78.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 11.5%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 11.5%

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

   5. Simplified 11.5%

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

   6. Taylor expanded in l around 0 10.3%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 10.3%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 10.3%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 10.3%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 9.8%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 9.8%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 9.8%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 9.8%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 9.8%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 17.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]

   if 5.49999999999999987e-126 < l

   1. Initial program 75.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 56.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in K around 0 72.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. cos-neg 72.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

   8. Simplified 72.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+147}:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(\frac{K \cdot m}{2} - M\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-127}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -1.9e+146)
   (*
    (cos (- (/ (* K (+ m n)) 2.0) M))
    (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))
   (if (<= l 6.2e-127)
     (* 0.5 (* K (* m (* (sin M) (- 1.0 l)))))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1.9e+146) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else if (l <= 6.2e-127) {
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-1.9d+146)) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (l <= 6.2d-127) then
        tmp = 0.5d0 * (k * (m * (sin(m_1) * (1.0d0 - l))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1.9e+146) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else if (l <= 6.2e-127) {
		tmp = 0.5 * (K * (m * (Math.sin(M) * (1.0 - l))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -1.9e+146:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))
	elif l <= 6.2e-127:
		tmp = 0.5 * (K * (m * (math.sin(M) * (1.0 - l))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -1.9e+146)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))));
	elseif (l <= 6.2e-127)
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * Float64(1.0 - l)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -1.9e+146)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	elseif (l <= 6.2e-127)
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -1.9e+146], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e-127], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.8999999999999999e146

   1. Initial program 81.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 40.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 40.1%

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

   5. Simplified 40.1%

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

   6. Taylor expanded in l around 0 40.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]

   if -1.8999999999999999e146 < l < 6.2e-127

   1. Initial program 78.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 11.5%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 11.5%

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

   5. Simplified 11.5%

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

   6. Taylor expanded in l around 0 10.3%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 10.3%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 10.3%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 10.3%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 9.8%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 9.8%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 9.8%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 9.8%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 9.8%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 17.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]

   if 6.2e-127 < l

   1. Initial program 75.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 56.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in K around 0 72.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. cos-neg 72.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

   8. Simplified 72.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-127}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (- m l) (+ n (pow (- (* (+ m n) 0.5) M) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((m - l) - (n + pow((((m + n) * 0.5) - M), 2.0))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(((m - l) - (n + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((m - l) - (n + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

def code(K, m, n, M, l):
	return math.exp(((m - l) - (n + math.pow((((m + n) * 0.5) - M), 2.0))))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(m - l) - Float64(n + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((m - l) - (n + ((((m + n) * 0.5) - M) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(m - l), $MachinePrecision] - N[(n + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\left(\ell - \left|m - n\right|\right)\right)}} \]

   2. distribute-neg-out 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)}} \]

   3. div-inv 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   4. metadata-eval 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   5. add-sqr-sqrt 36.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right|\right)\right)} \]

   6. fabs-sqr 36.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right)\right)} \]

   7. add-sqr-sqrt 77.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\left(m - n\right)}\right)\right)} \]

4. Applied egg-rr 77.6%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left(m - n\right)\right)\right)}} \]

5. Taylor expanded in K around 0 95.7%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{m - \left(\ell + \left(n + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)\right)}} \]

7. Simplified 95.7%

   \[\leadsto \color{blue}{\cos M \cdot e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}} \]

8. Taylor expanded in M around 0 94.8%

   \[\leadsto \color{blue}{1} \cdot e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \]

9. Final simplification 94.8%

   \[\leadsto e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \]
10. Add Preprocessing

Alternative 8: 86.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{m - \left(\left(n + \ell\right) + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- m (+ (+ n l) (* 0.25 (pow (+ m n) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((m - ((n + l) + (0.25 * pow((m + n), 2.0)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((m - ((n + l) + (0.25d0 * ((m + n) ** 2.0d0)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((m - ((n + l) + (0.25 * Math.pow((m + n), 2.0)))));
}

Python

def code(K, m, n, M, l):
	return math.exp((m - ((n + l) + (0.25 * math.pow((m + n), 2.0)))))

Julia

function code(K, m, n, M, l)
	return exp(Float64(m - Float64(Float64(n + l) + Float64(0.25 * (Float64(m + n) ^ 2.0)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((m - ((n + l) + (0.25 * ((m + n) ^ 2.0)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(m - N[(N[(n + l), $MachinePrecision] + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) + \left(-\left(\ell - \left|m - n\right|\right)\right)}} \]

   2. distribute-neg-out 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)}} \]

   3. div-inv 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   4. metadata-eval 77.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)} \]

   5. add-sqr-sqrt 36.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right|\right)\right)} \]

   6. fabs-sqr 36.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right)\right)} \]

   7. add-sqr-sqrt 77.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \color{blue}{\left(m - n\right)}\right)\right)} \]

4. Applied egg-rr 77.6%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \left(\ell - \left(m - n\right)\right)\right)}} \]

5. Taylor expanded in K around 0 95.7%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot e^{m - \left(\ell + \left(n + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)\right)}} \]

7. Simplified 95.7%

   \[\leadsto \color{blue}{\cos M \cdot e^{\left(m - \ell\right) - \left(n + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}} \]

8. Taylor expanded in M around 0 84.5%

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

   1. associate-+r+ 84.5%

      \[\leadsto e^{m - \color{blue}{\left(\left(\ell + n\right) + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]

   2. +-commutative 84.5%

      \[\leadsto e^{m - \left(\left(\ell + n\right) + 0.25 \cdot {\color{blue}{\left(n + m\right)}}^{2}\right)} \]

10. Simplified 84.5%

    \[\leadsto \color{blue}{e^{m - \left(\left(\ell + n\right) + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]

11. Final simplification 84.5%

    \[\leadsto e^{m - \left(\left(n + \ell\right) + 0.25 \cdot {\left(m + n\right)}^{2}\right)} \]
12. Add Preprocessing

Alternative 9: 17.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin M \cdot \left(1 - \ell\right)\\ \mathbf{if}\;n \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(n \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1.52 \cdot 10^{+32}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(K \cdot m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (sin M) (- 1.0 l))))
   (if (<= n 1.12e-169)
     (* (* K 0.5) (* n t_0))
     (if (<= n 1.52e+32)
       (*
        (cos (- (/ (* K (+ m n)) 2.0) M))
        (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))
       (* 0.5 (* t_0 (* K m)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = sin(M) * (1.0 - l);
	double tmp;
	if (n <= 1.12e-169) {
		tmp = (K * 0.5) * (n * t_0);
	} else if (n <= 1.52e+32) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 0.5 * (t_0 * (K * m));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(m_1) * (1.0d0 - l)
    if (n <= 1.12d-169) then
        tmp = (k * 0.5d0) * (n * t_0)
    else if (n <= 1.52d+32) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 0.5d0 * (t_0 * (k * m))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.sin(M) * (1.0 - l);
	double tmp;
	if (n <= 1.12e-169) {
		tmp = (K * 0.5) * (n * t_0);
	} else if (n <= 1.52e+32) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 0.5 * (t_0 * (K * m));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.sin(M) * (1.0 - l)
	tmp = 0
	if n <= 1.12e-169:
		tmp = (K * 0.5) * (n * t_0)
	elif n <= 1.52e+32:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 0.5 * (t_0 * (K * m))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(sin(M) * Float64(1.0 - l))
	tmp = 0.0
	if (n <= 1.12e-169)
		tmp = Float64(Float64(K * 0.5) * Float64(n * t_0));
	elseif (n <= 1.52e+32)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(0.5 * Float64(t_0 * Float64(K * m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = sin(M) * (1.0 - l);
	tmp = 0.0;
	if (n <= 1.12e-169)
		tmp = (K * 0.5) * (n * t_0);
	elseif (n <= 1.52e+32)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	else
		tmp = 0.5 * (t_0 * (K * m));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 1.12e-169], N[(N[(K * 0.5), $MachinePrecision] * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.52e+32], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 * N[(K * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.11999999999999998e-169

   1. Initial program 79.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 27.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 27.1%

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

   5. Simplified 27.1%

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

   6. Taylor expanded in l around 0 9.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 9.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 9.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 9.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 8.5%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 8.5%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 8.5%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 8.5%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 8.5%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in n around inf 27.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 27.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 27.7%

          \[\leadsto \left(0.5 \cdot K\right) \cdot \left(n \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 27.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(n \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]

   if 1.11999999999999998e-169 < n < 1.5200000000000001e32

   1. Initial program 86.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 43.5%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in l around 0 25.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]

   if 1.5200000000000001e32 < n

   1. Initial program 69.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 27.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 27.6%

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

   5. Simplified 27.6%

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

   6. Taylor expanded in l around 0 2.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 2.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 2.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 2.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 2.4%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 2.4%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 2.4%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 2.4%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 2.4%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 17.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 22.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(K \cdot m\right) \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 22.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(m \cdot K\right)} \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right) \]

       3. *-commutative 22.9%

          \[\leadsto 0.5 \cdot \left(\left(m \cdot K\right) \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 22.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\left(m \cdot K\right) \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.52 \cdot 10^{+32}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sin M \cdot \left(1 - \ell\right)\right) \cdot \left(K \cdot m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 17.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin M \cdot \left(1 - \ell\right)\\ \mathbf{if}\;n \leq 7.2 \cdot 10^{-170}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(n \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+32}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(K \cdot m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (sin M) (- 1.0 l))))
   (if (<= n 7.2e-170)
     (* (* K 0.5) (* n t_0))
     (if (<= n 1.3e+32)
       (* (cos (- (/ (* K (+ m n)) 2.0) M)) (+ 1.0 (* l (+ (* l 0.5) -1.0))))
       (* 0.5 (* t_0 (* K m)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = sin(M) * (1.0 - l);
	double tmp;
	if (n <= 7.2e-170) {
		tmp = (K * 0.5) * (n * t_0);
	} else if (n <= 1.3e+32) {
		tmp = cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	} else {
		tmp = 0.5 * (t_0 * (K * m));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(m_1) * (1.0d0 - l)
    if (n <= 7.2d-170) then
        tmp = (k * 0.5d0) * (n * t_0)
    else if (n <= 1.3d+32) then
        tmp = cos((((k * (m + n)) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
    else
        tmp = 0.5d0 * (t_0 * (k * m))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.sin(M) * (1.0 - l);
	double tmp;
	if (n <= 7.2e-170) {
		tmp = (K * 0.5) * (n * t_0);
	} else if (n <= 1.3e+32) {
		tmp = Math.cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	} else {
		tmp = 0.5 * (t_0 * (K * m));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.sin(M) * (1.0 - l)
	tmp = 0
	if n <= 7.2e-170:
		tmp = (K * 0.5) * (n * t_0)
	elif n <= 1.3e+32:
		tmp = math.cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)))
	else:
		tmp = 0.5 * (t_0 * (K * m))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(sin(M) * Float64(1.0 - l))
	tmp = 0.0
	if (n <= 7.2e-170)
		tmp = Float64(Float64(K * 0.5) * Float64(n * t_0));
	elseif (n <= 1.3e+32)
		tmp = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))));
	else
		tmp = Float64(0.5 * Float64(t_0 * Float64(K * m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = sin(M) * (1.0 - l);
	tmp = 0.0;
	if (n <= 7.2e-170)
		tmp = (K * 0.5) * (n * t_0);
	elseif (n <= 1.3e+32)
		tmp = cos((((K * (m + n)) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	else
		tmp = 0.5 * (t_0 * (K * m));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 7.2e-170], N[(N[(K * 0.5), $MachinePrecision] * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e+32], N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 * N[(K * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 7.2000000000000006e-170

   1. Initial program 79.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 27.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 27.1%

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

   5. Simplified 27.1%

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

   6. Taylor expanded in l around 0 9.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 9.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 9.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 9.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 8.5%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 8.5%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 8.5%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 8.5%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 8.5%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in n around inf 27.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 27.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 27.7%

          \[\leadsto \left(0.5 \cdot K\right) \cdot \left(n \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 27.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(n \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]

   if 7.2000000000000006e-170 < n < 1.3000000000000001e32

   1. Initial program 86.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 43.5%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in l around 0 25.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(0.5 \cdot \ell - 1\right)\right)} \]

   if 1.3000000000000001e32 < n

   1. Initial program 69.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 27.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 27.6%

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

   5. Simplified 27.6%

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

   6. Taylor expanded in l around 0 2.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 2.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 2.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 2.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 2.4%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 2.4%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 2.4%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 2.4%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 2.4%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 17.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 22.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(K \cdot m\right) \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 22.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(m \cdot K\right)} \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right) \]

       3. *-commutative 22.9%

          \[\leadsto 0.5 \cdot \left(\left(m \cdot K\right) \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 22.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\left(m \cdot K\right) \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{-170}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+32}:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sin M \cdot \left(1 - \ell\right)\right) \cdot \left(K \cdot m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-170} \lor \neg \left(n \leq 8.6 \cdot 10^{-49}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sin M \cdot \left(1 - \ell\right)\right) \cdot \left(K \cdot m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n 4.2e-170) (not (<= n 8.6e-49)))
   (* 0.5 (* (* (sin M) (- 1.0 l)) (* K m)))
   (cos (- (* (+ m n) (* K 0.5)) M))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= 4.2e-170) || !(n <= 8.6e-49)) {
		tmp = 0.5 * ((sin(M) * (1.0 - l)) * (K * m));
	} else {
		tmp = cos((((m + n) * (K * 0.5)) - M));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= 4.2d-170) .or. (.not. (n <= 8.6d-49))) then
        tmp = 0.5d0 * ((sin(m_1) * (1.0d0 - l)) * (k * m))
    else
        tmp = cos((((m + n) * (k * 0.5d0)) - m_1))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= 4.2e-170) || !(n <= 8.6e-49)) {
		tmp = 0.5 * ((Math.sin(M) * (1.0 - l)) * (K * m));
	} else {
		tmp = Math.cos((((m + n) * (K * 0.5)) - M));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= 4.2e-170) or not (n <= 8.6e-49):
		tmp = 0.5 * ((math.sin(M) * (1.0 - l)) * (K * m))
	else:
		tmp = math.cos((((m + n) * (K * 0.5)) - M))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= 4.2e-170) || !(n <= 8.6e-49))
		tmp = Float64(0.5 * Float64(Float64(sin(M) * Float64(1.0 - l)) * Float64(K * m)));
	else
		tmp = cos(Float64(Float64(Float64(m + n) * Float64(K * 0.5)) - M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= 4.2e-170) || ~((n <= 8.6e-49)))
		tmp = 0.5 * ((sin(M) * (1.0 - l)) * (K * m));
	else
		tmp = cos((((m + n) * (K * 0.5)) - M));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, 4.2e-170], N[Not[LessEqual[n, 8.6e-49]], $MachinePrecision]], N[(0.5 * N[(N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision] * N[(K * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cos[N[(N[(N[(m + n), $MachinePrecision] * N[(K * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 4.2000000000000001e-170 or 8.60000000000000033e-49 < n

   1. Initial program 77.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 28.9%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 28.9%

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

   5. Simplified 28.9%

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

   6. Taylor expanded in l around 0 6.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 6.6%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 6.6%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 6.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 6.8%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 6.8%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 6.8%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 6.8%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 6.8%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 15.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 17.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(K \cdot m\right) \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 17.5%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(m \cdot K\right)} \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right) \]

       3. *-commutative 17.5%

          \[\leadsto 0.5 \cdot \left(\left(m \cdot K\right) \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 17.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(\left(m \cdot K\right) \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]

   if 4.2000000000000001e-170 < n < 8.60000000000000033e-49

   1. Initial program 82.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 42.5%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 42.5%

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

   5. Simplified 42.5%

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

   6. Taylor expanded in l around 0 21.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 21.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 21.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 21.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in l around 0 21.1%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right)} \]
   10. Step-by-step derivation

       1. *-commutative 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(K \cdot \left(m + n\right)\right) \cdot 0.5} - M\right) \]

       2. *-commutative 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(\left(m + n\right) \cdot K\right)} \cdot 0.5 - M\right) \]

       3. associate-*l* 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(m + n\right) \cdot \left(K \cdot 0.5\right)} - M\right) \]

       4. +-commutative 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(n + m\right)} \cdot \left(K \cdot 0.5\right) - M\right) \]

       5. *-commutative 21.1%

          \[\leadsto \cos \left(\left(n + m\right) \cdot \color{blue}{\left(0.5 \cdot K\right)} - M\right) \]

   11. Simplified 21.1%

       \[\leadsto \color{blue}{\cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-170} \lor \neg \left(n \leq 8.6 \cdot 10^{-49}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sin M \cdot \left(1 - \ell\right)\right) \cdot \left(K \cdot m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 16.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin M \cdot \left(1 - \ell\right)\\ \mathbf{if}\;n \leq 1.16 \cdot 10^{-169}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(n \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(K \cdot m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (sin M) (- 1.0 l))))
   (if (<= n 1.16e-169)
     (* (* K 0.5) (* n t_0))
     (if (<= n 1.1e-46)
       (cos (- (* (+ m n) (* K 0.5)) M))
       (* 0.5 (* t_0 (* K m)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = sin(M) * (1.0 - l);
	double tmp;
	if (n <= 1.16e-169) {
		tmp = (K * 0.5) * (n * t_0);
	} else if (n <= 1.1e-46) {
		tmp = cos((((m + n) * (K * 0.5)) - M));
	} else {
		tmp = 0.5 * (t_0 * (K * m));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(m_1) * (1.0d0 - l)
    if (n <= 1.16d-169) then
        tmp = (k * 0.5d0) * (n * t_0)
    else if (n <= 1.1d-46) then
        tmp = cos((((m + n) * (k * 0.5d0)) - m_1))
    else
        tmp = 0.5d0 * (t_0 * (k * m))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.sin(M) * (1.0 - l);
	double tmp;
	if (n <= 1.16e-169) {
		tmp = (K * 0.5) * (n * t_0);
	} else if (n <= 1.1e-46) {
		tmp = Math.cos((((m + n) * (K * 0.5)) - M));
	} else {
		tmp = 0.5 * (t_0 * (K * m));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.sin(M) * (1.0 - l)
	tmp = 0
	if n <= 1.16e-169:
		tmp = (K * 0.5) * (n * t_0)
	elif n <= 1.1e-46:
		tmp = math.cos((((m + n) * (K * 0.5)) - M))
	else:
		tmp = 0.5 * (t_0 * (K * m))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(sin(M) * Float64(1.0 - l))
	tmp = 0.0
	if (n <= 1.16e-169)
		tmp = Float64(Float64(K * 0.5) * Float64(n * t_0));
	elseif (n <= 1.1e-46)
		tmp = cos(Float64(Float64(Float64(m + n) * Float64(K * 0.5)) - M));
	else
		tmp = Float64(0.5 * Float64(t_0 * Float64(K * m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = sin(M) * (1.0 - l);
	tmp = 0.0;
	if (n <= 1.16e-169)
		tmp = (K * 0.5) * (n * t_0);
	elseif (n <= 1.1e-46)
		tmp = cos((((m + n) * (K * 0.5)) - M));
	else
		tmp = 0.5 * (t_0 * (K * m));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 1.16e-169], N[(N[(K * 0.5), $MachinePrecision] * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1e-46], N[Cos[N[(N[(N[(m + n), $MachinePrecision] * N[(K * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(t$95$0 * N[(K * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.16e-169

   1. Initial program 79.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 27.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 27.1%

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

   5. Simplified 27.1%

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

   6. Taylor expanded in l around 0 9.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 9.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 9.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 9.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 8.5%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 8.5%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 8.5%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 8.5%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 8.5%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in n around inf 27.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 27.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 27.7%

          \[\leadsto \left(0.5 \cdot K\right) \cdot \left(n \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 27.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot K\right) \cdot \left(n \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]

   if 1.16e-169 < n < 1.1e-46

   1. Initial program 82.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 42.5%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 42.5%

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

   5. Simplified 42.5%

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

   6. Taylor expanded in l around 0 21.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 21.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 21.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 21.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in l around 0 21.1%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right)} \]
   10. Step-by-step derivation

       1. *-commutative 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(K \cdot \left(m + n\right)\right) \cdot 0.5} - M\right) \]

       2. *-commutative 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(\left(m + n\right) \cdot K\right)} \cdot 0.5 - M\right) \]

       3. associate-*l* 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(m + n\right) \cdot \left(K \cdot 0.5\right)} - M\right) \]

       4. +-commutative 21.1%

          \[\leadsto \cos \left(\color{blue}{\left(n + m\right)} \cdot \left(K \cdot 0.5\right) - M\right) \]

       5. *-commutative 21.1%

          \[\leadsto \cos \left(\left(n + m\right) \cdot \color{blue}{\left(0.5 \cdot K\right)} - M\right) \]

   11. Simplified 21.1%

       \[\leadsto \color{blue}{\cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)} \]

   if 1.1e-46 < n

   1. Initial program 74.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 31.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 31.8%

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

   5. Simplified 31.8%

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

   6. Taylor expanded in l around 0 2.9%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 2.9%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 2.9%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 2.9%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 4.2%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 4.2%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 4.2%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 4.2%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 4.2%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 17.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 22.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(K \cdot m\right) \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)} \]

       2. *-commutative 22.3%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(m \cdot K\right)} \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right) \]

       3. *-commutative 22.3%

          \[\leadsto 0.5 \cdot \left(\left(m \cdot K\right) \cdot \color{blue}{\left(\left(1 - \ell\right) \cdot \sin M\right)}\right) \]

   14. Simplified 22.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(\left(m \cdot K\right) \cdot \left(\left(1 - \ell\right) \cdot \sin M\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.16 \cdot 10^{-169}:\\ \;\;\;\;\left(K \cdot 0.5\right) \cdot \left(n \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sin M \cdot \left(1 - \ell\right)\right) \cdot \left(K \cdot m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 10.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.4 \cdot 10^{-48}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 3.4e-48)
   (cos (- (* (+ m n) (* K 0.5)) M))
   (* 0.5 (* K (* m (* (sin M) (- 1.0 l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3.4e-48) {
		tmp = cos((((m + n) * (K * 0.5)) - M));
	} else {
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 3.4d-48) then
        tmp = cos((((m + n) * (k * 0.5d0)) - m_1))
    else
        tmp = 0.5d0 * (k * (m * (sin(m_1) * (1.0d0 - l))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3.4e-48) {
		tmp = Math.cos((((m + n) * (K * 0.5)) - M));
	} else {
		tmp = 0.5 * (K * (m * (Math.sin(M) * (1.0 - l))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 3.4e-48:
		tmp = math.cos((((m + n) * (K * 0.5)) - M))
	else:
		tmp = 0.5 * (K * (m * (math.sin(M) * (1.0 - l))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 3.4e-48)
		tmp = cos(Float64(Float64(Float64(m + n) * Float64(K * 0.5)) - M));
	else
		tmp = Float64(0.5 * Float64(K * Float64(m * Float64(sin(M) * Float64(1.0 - l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 3.4e-48)
		tmp = cos((((m + n) * (K * 0.5)) - M));
	else
		tmp = 0.5 * (K * (m * (sin(M) * (1.0 - l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 3.4e-48], N[Cos[N[(N[(N[(m + n), $MachinePrecision] * N[(K * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(K * N[(m * N[(N[Sin[M], $MachinePrecision] * N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.40000000000000028e-48

   1. Initial program 79.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 29.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 29.6%

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

   5. Simplified 29.6%

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

   6. Taylor expanded in l around 0 11.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 11.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 11.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 11.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in l around 0 11.1%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right)} \]
   10. Step-by-step derivation

       1. *-commutative 11.1%

          \[\leadsto \cos \left(\color{blue}{\left(K \cdot \left(m + n\right)\right) \cdot 0.5} - M\right) \]

       2. *-commutative 11.1%

          \[\leadsto \cos \left(\color{blue}{\left(\left(m + n\right) \cdot K\right)} \cdot 0.5 - M\right) \]

       3. associate-*l* 11.1%

          \[\leadsto \cos \left(\color{blue}{\left(m + n\right) \cdot \left(K \cdot 0.5\right)} - M\right) \]

       4. +-commutative 11.1%

          \[\leadsto \cos \left(\color{blue}{\left(n + m\right)} \cdot \left(K \cdot 0.5\right) - M\right) \]

       5. *-commutative 11.1%

          \[\leadsto \cos \left(\left(n + m\right) \cdot \color{blue}{\left(0.5 \cdot K\right)} - M\right) \]

   11. Simplified 11.1%

       \[\leadsto \color{blue}{\cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)} \]

   if 3.40000000000000028e-48 < n

   1. Initial program 74.4%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf 31.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg 31.8%

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

   5. Simplified 31.8%

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

   6. Taylor expanded in l around 0 2.9%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 2.9%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

      2. unsub-neg 2.9%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   8. Simplified 2.9%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

   9. Taylor expanded in K around 0 4.2%

      \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
   10. Step-by-step derivation

       1. cos-neg 4.2%

          \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

       2. associate-*r* 4.2%

          \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

       3. sin-neg 4.2%

          \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

   11. Simplified 4.2%

       \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

   12. Taylor expanded in m around inf 17.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.4 \cdot 10^{-48}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \left(\sin M \cdot \left(1 - \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 6.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \left(1 - \ell\right) \cdot \left(1 + M \cdot \left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5 + M \cdot \left(-0.08333333333333333 \cdot \left(K \cdot \left(\left(m + n\right) \cdot M\right)\right) - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (- 1.0 l)
  (+
   1.0
   (*
    M
    (+
     (* (* K (+ m n)) 0.5)
     (* M (- (* -0.08333333333333333 (* K (* (+ m n) M))) 0.5)))))))

C

double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * ((-0.08333333333333333 * (K * ((m + n) * M))) - 0.5)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = (1.0d0 - l) * (1.0d0 + (m_1 * (((k * (m + n)) * 0.5d0) + (m_1 * (((-0.08333333333333333d0) * (k * ((m + n) * m_1))) - 0.5d0)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * ((-0.08333333333333333 * (K * ((m + n) * M))) - 0.5)))));
}

Python

def code(K, m, n, M, l):
	return (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * ((-0.08333333333333333 * (K * ((m + n) * M))) - 0.5)))))

Julia

function code(K, m, n, M, l)
	return Float64(Float64(1.0 - l) * Float64(1.0 + Float64(M * Float64(Float64(Float64(K * Float64(m + n)) * 0.5) + Float64(M * Float64(Float64(-0.08333333333333333 * Float64(K * Float64(Float64(m + n) * M))) - 0.5))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * ((-0.08333333333333333 * (K * ((m + n) * M))) - 0.5)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(1.0 - l), $MachinePrecision] * N[(1.0 + N[(M * N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(M * N[(N[(-0.08333333333333333 * N[(K * N[(N[(m + n), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in l around inf 30.4%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
4. Step-by-step derivation

   1. mul-1-neg 30.4%

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

5. Simplified 30.4%

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

6. Taylor expanded in l around 0 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
7. Step-by-step derivation

   1. neg-mul-1 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

   2. unsub-neg 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

8. Simplified 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

9. Taylor expanded in K around 0 8.6%

   \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
10. Step-by-step derivation

    1. cos-neg 8.6%

       \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

    2. associate-*r* 8.6%

       \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

    3. sin-neg 8.6%

       \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

11. Simplified 8.6%

    \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

12. Taylor expanded in M around 0 8.9%

    \[\leadsto \color{blue}{\left(1 + M \cdot \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) + M \cdot \left(-0.08333333333333333 \cdot \left(K \cdot \left(M \cdot \left(m + n\right)\right)\right) - 0.5\right)\right)\right)} \cdot \left(1 - \ell\right) \]

13. Final simplification 8.9%

    \[\leadsto \left(1 - \ell\right) \cdot \left(1 + M \cdot \left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5 + M \cdot \left(-0.08333333333333333 \cdot \left(K \cdot \left(\left(m + n\right) \cdot M\right)\right) - 0.5\right)\right)\right) \]
14. Add Preprocessing

Alternative 15: 6.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \left(1 - \ell\right) \cdot \left(1 + M \cdot \left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5 + M \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (- 1.0 l) (+ 1.0 (* M (+ (* (* K (+ m n)) 0.5) (* M -0.5))))))

C

double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * -0.5))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = (1.0d0 - l) * (1.0d0 + (m_1 * (((k * (m + n)) * 0.5d0) + (m_1 * (-0.5d0)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * -0.5))));
}

Python

def code(K, m, n, M, l):
	return (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * -0.5))))

Julia

function code(K, m, n, M, l)
	return Float64(Float64(1.0 - l) * Float64(1.0 + Float64(M * Float64(Float64(Float64(K * Float64(m + n)) * 0.5) + Float64(M * -0.5)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = (1.0 - l) * (1.0 + (M * (((K * (m + n)) * 0.5) + (M * -0.5))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(1.0 - l), $MachinePrecision] * N[(1.0 + N[(M * N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in l around inf 30.4%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
4. Step-by-step derivation

   1. mul-1-neg 30.4%

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

5. Simplified 30.4%

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

6. Taylor expanded in l around 0 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
7. Step-by-step derivation

   1. neg-mul-1 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

   2. unsub-neg 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

8. Simplified 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

9. Taylor expanded in K around 0 8.6%

   \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
10. Step-by-step derivation

    1. cos-neg 8.6%

       \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

    2. associate-*r* 8.6%

       \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

    3. sin-neg 8.6%

       \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

11. Simplified 8.6%

    \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

12. Taylor expanded in M around 0 8.8%

    \[\leadsto \color{blue}{\left(1 + M \cdot \left(-0.5 \cdot M + 0.5 \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]

13. Final simplification 8.8%

    \[\leadsto \left(1 - \ell\right) \cdot \left(1 + M \cdot \left(\left(K \cdot \left(m + n\right)\right) \cdot 0.5 + M \cdot -0.5\right)\right) \]
14. Add Preprocessing

Alternative 16: 6.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \left(1 - \ell\right) \cdot \left(1 + 0.5 \cdot \left(\left(m + n\right) \cdot \left(K \cdot M\right)\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (- 1.0 l) (+ 1.0 (* 0.5 (* (+ m n) (* K M))))))

C

double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (0.5 * ((m + n) * (K * M))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = (1.0d0 - l) * (1.0d0 + (0.5d0 * ((m + n) * (k * m_1))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (0.5 * ((m + n) * (K * M))));
}

Python

def code(K, m, n, M, l):
	return (1.0 - l) * (1.0 + (0.5 * ((m + n) * (K * M))))

Julia

function code(K, m, n, M, l)
	return Float64(Float64(1.0 - l) * Float64(1.0 + Float64(0.5 * Float64(Float64(m + n) * Float64(K * M)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = (1.0 - l) * (1.0 + (0.5 * ((m + n) * (K * M))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(1.0 - l), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(N[(m + n), $MachinePrecision] * N[(K * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in l around inf 30.4%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
4. Step-by-step derivation

   1. mul-1-neg 30.4%

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

5. Simplified 30.4%

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

6. Taylor expanded in l around 0 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
7. Step-by-step derivation

   1. neg-mul-1 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

   2. unsub-neg 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

8. Simplified 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

9. Taylor expanded in K around 0 8.6%

   \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
10. Step-by-step derivation

    1. cos-neg 8.6%

       \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

    2. associate-*r* 8.6%

       \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

    3. sin-neg 8.6%

       \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

11. Simplified 8.6%

    \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

12. Taylor expanded in M around 0 8.4%

    \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(K \cdot \left(M \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
13. Step-by-step derivation

    1. associate-*r* 8.4%

       \[\leadsto \left(1 + 0.5 \cdot \color{blue}{\left(\left(K \cdot M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

    2. +-commutative 8.4%

       \[\leadsto \left(1 + 0.5 \cdot \left(\left(K \cdot M\right) \cdot \color{blue}{\left(n + m\right)}\right)\right) \cdot \left(1 - \ell\right) \]

14. Simplified 8.4%

    \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(\left(K \cdot M\right) \cdot \left(n + m\right)\right)\right)} \cdot \left(1 - \ell\right) \]

15. Final simplification 8.4%

    \[\leadsto \left(1 - \ell\right) \cdot \left(1 + 0.5 \cdot \left(\left(m + n\right) \cdot \left(K \cdot M\right)\right)\right) \]
16. Add Preprocessing

Alternative 17: 6.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \left(1 - \ell\right) \cdot \left(1 + 0.5 \cdot \left(K \cdot \left(\left(m + n\right) \cdot M\right)\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (- 1.0 l) (+ 1.0 (* 0.5 (* K (* (+ m n) M))))))

C

double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (0.5 * (K * ((m + n) * M))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = (1.0d0 - l) * (1.0d0 + (0.5d0 * (k * ((m + n) * m_1))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return (1.0 - l) * (1.0 + (0.5 * (K * ((m + n) * M))));
}

Python

def code(K, m, n, M, l):
	return (1.0 - l) * (1.0 + (0.5 * (K * ((m + n) * M))))

Julia

function code(K, m, n, M, l)
	return Float64(Float64(1.0 - l) * Float64(1.0 + Float64(0.5 * Float64(K * Float64(Float64(m + n) * M)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = (1.0 - l) * (1.0 + (0.5 * (K * ((m + n) * M))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[(1.0 - l), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(K * N[(N[(m + n), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in l around inf 30.4%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
4. Step-by-step derivation

   1. mul-1-neg 30.4%

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

5. Simplified 30.4%

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

6. Taylor expanded in l around 0 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
7. Step-by-step derivation

   1. neg-mul-1 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

   2. unsub-neg 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

8. Simplified 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

9. Taylor expanded in K around 0 8.6%

   \[\leadsto \color{blue}{\left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]
10. Step-by-step derivation

    1. cos-neg 8.6%

       \[\leadsto \left(\color{blue}{\cos M} + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot \left(1 - \ell\right) \]

    2. associate-*r* 8.6%

       \[\leadsto \left(\cos M + \color{blue}{\left(-0.5 \cdot K\right) \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)}\right) \cdot \left(1 - \ell\right) \]

    3. sin-neg 8.6%

       \[\leadsto \left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\color{blue}{\left(-\sin M\right)} \cdot \left(m + n\right)\right)\right) \cdot \left(1 - \ell\right) \]

11. Simplified 8.6%

    \[\leadsto \color{blue}{\left(\cos M + \left(-0.5 \cdot K\right) \cdot \left(\left(-\sin M\right) \cdot \left(m + n\right)\right)\right)} \cdot \left(1 - \ell\right) \]

12. Taylor expanded in M around 0 8.4%

    \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(K \cdot \left(M \cdot \left(m + n\right)\right)\right)\right)} \cdot \left(1 - \ell\right) \]

13. Final simplification 8.4%

    \[\leadsto \left(1 - \ell\right) \cdot \left(1 + 0.5 \cdot \left(K \cdot \left(\left(m + n\right) \cdot M\right)\right)\right) \]
14. Add Preprocessing

Alternative 18: 6.8% accurate, 141.7× speedup

Math

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

FPCore

(FPCore (K m n M l) :precision binary64 (- 1.0 l))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 - l;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 - l
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 - l;
}

Python

def code(K, m, n, M, l):
	return 1.0 - l

Julia

function code(K, m, n, M, l)
	return Float64(1.0 - l)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 - l;
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 - l), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in l around inf 30.4%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
4. Step-by-step derivation

   1. mul-1-neg 30.4%

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

5. Simplified 30.4%

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

6. Taylor expanded in l around 0 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
7. Step-by-step derivation

   1. neg-mul-1 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(1 + \color{blue}{\left(-\ell\right)}\right) \]

   2. unsub-neg 8.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

8. Simplified 8.2%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

9. Taylor expanded in K around 0 8.3%

   \[\leadsto \color{blue}{\cos \left(-M\right) \cdot \left(1 - \ell\right)} \]
10. Step-by-step derivation

    1. cos-neg 8.3%

       \[\leadsto \color{blue}{\cos M} \cdot \left(1 - \ell\right) \]

11. Simplified 8.3%

    \[\leadsto \color{blue}{\cos M \cdot \left(1 - \ell\right)} \]

12. Taylor expanded in M around 0 8.3%

    \[\leadsto \color{blue}{1 - \ell} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))