Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.9%

Time: 15.4s

Alternatives: 11

Speedup: 1.9×

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: 75.7% 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: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_1 := \left(m + n\right) \cdot 0.5\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot \cos \left({\left(\sqrt[3]{\left(m + n\right) \cdot \left(K \cdot 0.5\right)}\right)}^{3} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) + \left(t\_1 - M\right) \cdot \left(M - t\_1\right)} \cdot \cos M\\ \end{array} \end{array} \]

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 (* (+ m n) 0.5)))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) INFINITY)
     (* t_0 (cos (- (pow (cbrt (* (+ m n) (* K 0.5))) 3.0) M)))
     (* (exp (+ (- n m) (* (- t_1 M) (- M t_1)))) (cos M)))))

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 = (m + n) * 0.5;
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= ((double) INFINITY)) {
		tmp = t_0 * cos((pow(cbrt(((m + n) * (K * 0.5))), 3.0) - M));
	} else {
		tmp = exp(((n - m) + ((t_1 - M) * (M - t_1)))) * cos(M);
	}
	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 = (m + n) * 0.5;
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.cos((Math.pow(Math.cbrt(((m + n) * (K * 0.5))), 3.0) - M));
	} else {
		tmp = Math.exp(((n - m) + ((t_1 - M) * (M - t_1)))) * Math.cos(M);
	}
	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(Float64(m + n) * 0.5)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = Float64(t_0 * cos(Float64((cbrt(Float64(Float64(m + n) * Float64(K * 0.5))) ^ 3.0) - M)));
	else
		tmp = Float64(exp(Float64(Float64(n - m) + Float64(Float64(t_1 - M) * Float64(M - t_1)))) * cos(M));
	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[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(t$95$0 * N[Cos[N[(N[Power[N[Power[N[(N[(m + n), $MachinePrecision] * N[(K * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(t$95$1 - M), $MachinePrecision] * N[(M - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $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)))))) < +inf.0

   1. Initial program 95.2%

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

      1. add-cube-cbrt 95.6%

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

      2. pow3 96.1%

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

      3. div-inv 96.1%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(K \cdot \left(m + n\right)\right) \cdot \frac{1}{2}}}\right)}^{3} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 96.1%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(\left(m + n\right) \cdot K\right)} \cdot \frac{1}{2}}\right)}^{3} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. associate-*l* 96.1%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(m + n\right) \cdot \left(K \cdot \frac{1}{2}\right)}}\right)}^{3} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      6. metadata-eval 96.1%

         \[\leadsto \cos \left({\left(\sqrt[3]{\left(m + n\right) \cdot \left(K \cdot \color{blue}{0.5}\right)}\right)}^{3} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   4. Applied egg-rr 96.1%

      \[\leadsto \cos \left(\color{blue}{{\left(\sqrt[3]{\left(m + n\right) \cdot \left(K \cdot 0.5\right)}\right)}^{3}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in K around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. rem-square-sqrt 54.3%

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

      2. fabs-sqr 54.3%

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

      3. rem-square-sqrt 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}}} \cdot \cos M \]
   9. Step-by-step derivation

      1. unpow2 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left({\left(\sqrt[3]{\left(m + n\right) \cdot \left(K \cdot 0.5\right)}\right)}^{3} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(M - \left(m + n\right) \cdot 0.5\right)} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<=
        (*
         (cos (- (/ (* K (+ m n)) 2.0) M))
         (exp (- (- t_0 l) (pow (- (/ (+ m n) 2.0) M) 2.0))))
        -0.5)
     (*
      (cos (- (/ (* K m) 2.0) M))
      (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))
     (* (cos M) (exp (- t_0 (+ l (pow (- (* (+ m n) 0.5) M) 2.0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * exp(((t_0 - l) - pow((((m + n) / 2.0) - M), 2.0)))) <= -0.5) {
		tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = cos(M) * exp((t_0 - (l + pow((((m + n) * 0.5) - M), 2.0))));
	}
	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 = abs((m - n))
    if ((cos((((k * (m + n)) / 2.0d0) - m_1)) * exp(((t_0 - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))) <= (-0.5d0)) then
        tmp = cos((((k * m) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = cos(m_1) * exp((t_0 - (l + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
    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.abs((m - n));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp(((t_0 - l) - Math.pow((((m + n) / 2.0) - M), 2.0)))) <= -0.5) {
		tmp = Math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * math.exp(((t_0 - l) - math.pow((((m + n) / 2.0) - M), 2.0)))) <= -0.5:
		tmp = math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - (l + math.pow((((m + n) * 0.5) - M), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(t_0 - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))) <= -0.5)
		tmp = Float64(cos(Float64(Float64(Float64(K * m) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * exp(((t_0 - l) - ((((m + n) / 2.0) - M) ^ 2.0)))) <= -0.5)
		tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	else
		tmp = cos(M) * exp((t_0 - (l + ((((m + n) * 0.5) - M) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[Cos[N[(N[(N[(K * m), $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[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $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)))))) < -0.5

   1. Initial program 54.5%

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

   3. Taylor expanded in l around inf 54.5%

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

      1. mul-1-neg 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in m around inf 37.6%

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

      1. *-commutative 37.6%

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

   8. Simplified 37.6%

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

   9. Taylor expanded in l around 0 37.6%

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

   if -0.5 < (*.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 79.4%

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

   3. Taylor expanded in K around 0 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 94.7%

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

Alternative 3: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+235}:\\ \;\;\;\;\cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 740:\\ \;\;\;\;e^{\left(n - m\right) + \left(t\_0 - M\right) \cdot \left(M - t\_0\right)} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (+ m n) 0.5)))
   (if (<= l -4.2e+235)
     (* (cos (- (/ (* K m) 2.0) M)) (+ 1.0 (* l (+ (* l 0.5) -1.0))))
     (if (<= l 740.0)
       (* (exp (+ (- n m) (* (- t_0 M) (- M t_0)))) (cos M))
       (* (cos M) (exp (- l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * 0.5;
	double tmp;
	if (l <= -4.2e+235) {
		tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	} else if (l <= 740.0) {
		tmp = exp(((n - m) + ((t_0 - M) * (M - t_0)))) * cos(M);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (m + n) * 0.5d0
    if (l <= (-4.2d+235)) then
        tmp = cos((((k * m) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
    else if (l <= 740.0d0) then
        tmp = exp(((n - m) + ((t_0 - m_1) * (m_1 - t_0)))) * cos(m_1)
    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 t_0 = (m + n) * 0.5;
	double tmp;
	if (l <= -4.2e+235) {
		tmp = Math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	} else if (l <= 740.0) {
		tmp = Math.exp(((n - m) + ((t_0 - M) * (M - t_0)))) * Math.cos(M);
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m + n) * 0.5
	tmp = 0
	if l <= -4.2e+235:
		tmp = math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)))
	elif l <= 740.0:
		tmp = math.exp(((n - m) + ((t_0 - M) * (M - t_0)))) * math.cos(M)
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * 0.5)
	tmp = 0.0
	if (l <= -4.2e+235)
		tmp = Float64(cos(Float64(Float64(Float64(K * m) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))));
	elseif (l <= 740.0)
		tmp = Float64(exp(Float64(Float64(n - m) + Float64(Float64(t_0 - M) * Float64(M - t_0)))) * cos(M));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m + n) * 0.5;
	tmp = 0.0;
	if (l <= -4.2e+235)
		tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
	elseif (l <= 740.0)
		tmp = exp(((n - m) + ((t_0 - M) * (M - t_0)))) * cos(M);
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[l, -4.2e+235], N[(N[Cos[N[(N[(N[(K * m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 740.0], N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.2000000000000001e235

   1. Initial program 78.3%

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

   3. Taylor expanded in l around inf 56.9%

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

      1. mul-1-neg 56.9%

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

   5. Simplified 56.9%

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

   6. Taylor expanded in m around inf 43.9%

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

      1. *-commutative 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in l around 0 43.9%

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

   if -4.2000000000000001e235 < l < 740

   1. Initial program 73.3%

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

   3. Taylor expanded in K around 0 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in l around 0 92.6%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. rem-square-sqrt 44.4%

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

      2. fabs-sqr 44.4%

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

      3. rem-square-sqrt 92.6%

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

   8. Simplified 92.6%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}}} \cdot \cos M \]
   9. Step-by-step derivation

      1. unpow2 92.6%

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

      2. +-commutative 92.6%

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

      3. +-commutative 92.6%

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

   10. Applied egg-rr 92.6%

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

   if 740 < l

   1. Initial program 90.6%

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

   3. Taylor expanded in l around inf 90.6%

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

      1. mul-1-neg 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+235}:\\ \;\;\;\;\cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 740:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(M - \left(m + n\right) \cdot 0.5\right)} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 34:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) + \left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 34.0)
   (* (cos M) (exp (+ (- n m) (* (- (* (+ m n) 0.5) M) (- M (* m 0.5))))))
   (exp (* -0.25 (pow n 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 34.0) {
		tmp = cos(M) * exp(((n - m) + ((((m + n) * 0.5) - M) * (M - (m * 0.5)))));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	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) :: tmp
    if (n <= 34.0d0) then
        tmp = cos(m_1) * exp(((n - m) + ((((m + n) * 0.5d0) - m_1) * (m_1 - (m * 0.5d0)))))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    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 <= 34.0) {
		tmp = Math.cos(M) * Math.exp(((n - m) + ((((m + n) * 0.5) - M) * (M - (m * 0.5)))));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 34.0:
		tmp = math.cos(M) * math.exp(((n - m) + ((((m + n) * 0.5) - M) * (M - (m * 0.5)))))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 34.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(n - m) + Float64(Float64(Float64(Float64(m + n) * 0.5) - M) * Float64(M - Float64(m * 0.5))))));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 34.0)
		tmp = cos(M) * exp(((n - m) + ((((m + n) * 0.5) - M) * (M - (m * 0.5)))));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 34.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 34

   1. Initial program 78.8%

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

   3. Taylor expanded in K around 0 93.4%

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

   5. Simplified 93.4%

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

   6. Taylor expanded in l around 0 81.7%

      \[\leadsto \color{blue}{e^{\left|n - m\right| - {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}}} \cdot \cos M \]
   7. Step-by-step derivation

      1. rem-square-sqrt 30.9%

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

      2. fabs-sqr 30.9%

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

      3. rem-square-sqrt 81.7%

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

   8. Simplified 81.7%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}}} \cdot \cos M \]
   9. Step-by-step derivation

      1. unpow2 81.7%

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

      2. +-commutative 81.7%

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

      3. +-commutative 81.7%

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

   10. Applied egg-rr 81.7%

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

   11. Taylor expanded in n around 0 78.0%

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

   if 34 < n

   1. Initial program 75.0%

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

   3. Taylor expanded in n around inf 75.0%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 34:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) + \left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K \cdot m}{2} - M\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+235}:\\ \;\;\;\;t\_0 \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{+32}:\\ \;\;\;\;t\_0 \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 700:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (/ (* K m) 2.0) M))))
   (if (<= l -4.2e+235)
     (* t_0 (+ 1.0 (* l (+ (* l 0.5) -1.0))))
     (if (<= l -1.55e+32)
       (* t_0 (exp l))
       (if (<= l 700.0)
         (* (cos M) (exp (* -0.25 (* n n))))
         (* (cos M) (exp (- l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos((((K * m) / 2.0) - M));
	double tmp;
	if (l <= -4.2e+235) {
		tmp = t_0 * (1.0 + (l * ((l * 0.5) + -1.0)));
	} else if (l <= -1.55e+32) {
		tmp = t_0 * exp(l);
	} else if (l <= 700.0) {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = cos((((k * m) / 2.0d0) - m_1))
    if (l <= (-4.2d+235)) then
        tmp = t_0 * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
    else if (l <= (-1.55d+32)) then
        tmp = t_0 * exp(l)
    else if (l <= 700.0d0) then
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    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 t_0 = Math.cos((((K * m) / 2.0) - M));
	double tmp;
	if (l <= -4.2e+235) {
		tmp = t_0 * (1.0 + (l * ((l * 0.5) + -1.0)));
	} else if (l <= -1.55e+32) {
		tmp = t_0 * Math.exp(l);
	} else if (l <= 700.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos((((K * m) / 2.0) - M))
	tmp = 0
	if l <= -4.2e+235:
		tmp = t_0 * (1.0 + (l * ((l * 0.5) + -1.0)))
	elif l <= -1.55e+32:
		tmp = t_0 * math.exp(l)
	elif l <= 700.0:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(Float64(K * m) / 2.0) - M))
	tmp = 0.0
	if (l <= -4.2e+235)
		tmp = Float64(t_0 * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))));
	elseif (l <= -1.55e+32)
		tmp = Float64(t_0 * exp(l));
	elseif (l <= 700.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos((((K * m) / 2.0) - M));
	tmp = 0.0;
	if (l <= -4.2e+235)
		tmp = t_0 * (1.0 + (l * ((l * 0.5) + -1.0)));
	elseif (l <= -1.55e+32)
		tmp = t_0 * exp(l);
	elseif (l <= 700.0)
		tmp = cos(M) * exp((-0.25 * (n * n)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(N[(K * m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.2e+235], N[(t$95$0 * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.55e+32], N[(t$95$0 * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 700.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.2000000000000001e235

   1. Initial program 78.3%

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

   3. Taylor expanded in l around inf 56.9%

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

      1. mul-1-neg 56.9%

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

   5. Simplified 56.9%

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

   6. Taylor expanded in m around inf 43.9%

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

      1. *-commutative 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in l around 0 43.9%

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

   if -4.2000000000000001e235 < l < -1.54999999999999997e32

   1. Initial program 68.8%

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

   3. Taylor expanded in l around inf 9.3%

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

      1. mul-1-neg 9.3%

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

   5. Simplified 9.3%

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

   6. Taylor expanded in m around inf 9.4%

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

      1. *-commutative 9.4%

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

   8. Simplified 9.4%

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

      1. pow1 9.4%

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

      2. associate-/l* 9.4%

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

      3. add-sqr-sqrt 9.4%

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

      4. sqrt-unprod 9.4%

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

      5. sqr-neg 9.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 66.9%

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

   10. Applied egg-rr 66.9%

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

       1. unpow1 66.9%

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

       2. associate-*r/ 66.9%

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

       3. *-commutative 66.9%

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

   12. Simplified 66.9%

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

   if -1.54999999999999997e32 < l < 700

   1. Initial program 75.1%

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

   3. Taylor expanded in n around inf 42.2%

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

   4. Taylor expanded in K around 0 56.8%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg 56.8%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 56.8%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 56.8%

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   8. Applied egg-rr 56.8%

      \[\leadsto \cos M \cdot e^{-0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   if 700 < l

   1. Initial program 90.6%

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

   3. Taylor expanded in l around inf 90.6%

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

      1. mul-1-neg 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+235}:\\ \;\;\;\;\cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right)\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{+32}:\\ \;\;\;\;\cos \left(\frac{K \cdot m}{2} - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 700:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -54.0)
   (exp (* -0.25 (pow n 2.0)))
   (if (<= n 55.0)
     (* (cos M) (exp (- l)))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -54.0) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else if (n <= 55.0) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	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 <= (-54.0d0)) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else if (n <= 55.0d0) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    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 <= -54.0) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else if (n <= 55.0) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -54.0:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	elif n <= 55.0:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -54.0)
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	elseif (n <= 55.0)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -54.0)
		tmp = exp((-0.25 * (n ^ 2.0)));
	elseif (n <= 55.0)
		tmp = cos(M) * exp(-l);
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -54.0], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -54

   1. Initial program 71.7%

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

   3. Taylor expanded in n around inf 68.4%

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

   4. Taylor expanded in K around 0 96.7%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg 96.7%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in M around 0 96.7%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]

   if -54 < n < 55

   1. Initial program 81.6%

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

   3. Taylor expanded in l around inf 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 K around 0 44.0%

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

      1. cos-neg 44.0%

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

      2. *-commutative 44.0%

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

   8. Simplified 44.0%

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

   if 55 < n

   1. Initial program 75.0%

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

   3. Taylor expanded in n around inf 75.0%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \color{blue}{\left(n \cdot n\right)}} \]

   8. Applied egg-rr 100.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 53\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -54.0) (not (<= n 53.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 53.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} 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 ((n <= (-54.0d0)) .or. (.not. (n <= 53.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    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 ((n <= -54.0) || !(n <= 53.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -54.0) or not (n <= 53.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -54.0) || !(n <= 53.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	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 ((n <= -54.0) || ~((n <= 53.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -54.0], N[Not[LessEqual[n, 53.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 53 < n

   1. Initial program 73.1%

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

   3. Taylor expanded in n around inf 71.3%

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

   4. Taylor expanded in K around 0 98.2%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg 98.2%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 98.2%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in M around 0 98.2%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]

   if -54 < n < 53

   1. Initial program 81.6%

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

   3. Taylor expanded in l around inf 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 K around 0 44.0%

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

      1. cos-neg 44.0%

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

      2. *-commutative 44.0%

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

   8. Simplified 44.0%

      \[\leadsto \color{blue}{e^{-\ell} \cdot \cos M} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 53\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-134} \lor \neg \left(n \leq 51\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -4.8e-134) (not (<= n 51.0)))
   (exp (* -0.25 (pow n 2.0)))
   (*
    (cos (- (/ (* K m) 2.0) M))
    (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -4.8e-134) || !(n <= 51.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	}
	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) :: tmp
    if ((n <= (-4.8d-134)) .or. (.not. (n <= 51.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos((((k * m) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= -4.8e-134) || !(n <= 51.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -4.8e-134) or not (n <= 51.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -4.8e-134) || !(n <= 51.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(Float64(Float64(Float64(K * m) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -4.8e-134) || ~((n <= 51.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -4.8e-134], N[Not[LessEqual[n, 51.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(N[(K * m), $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]]

Derivation

1. Split input into 2 regimes

2. if n < -4.80000000000000019e-134 or 51 < n

   1. Initial program 77.4%

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

   3. Taylor expanded in n around inf 55.3%

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

   4. Taylor expanded in K around 0 75.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{-0.25 \cdot {n}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg 75.3%

         \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   6. Simplified 75.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-0.25 \cdot {n}^{2}} \]

   7. Taylor expanded in M around 0 75.3%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]

   if -4.80000000000000019e-134 < n < 51

   1. Initial program 79.0%

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

   3. Taylor expanded in l around inf 43.0%

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

      1. mul-1-neg 43.0%

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

   5. Simplified 43.0%

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

   6. Taylor expanded in m around inf 41.8%

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

      1. *-commutative 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in l around 0 20.6%

      \[\leadsto \cos \left(\frac{m \cdot K}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-134} \lor \neg \left(n \leq 51\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K m) 2.0) M))
  (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
}

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((((k * m) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * m) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * m), $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]

Derivation

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 36.2%

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

   1. mul-1-neg 36.2%

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

5. Simplified 36.2%

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

6. Taylor expanded in m around inf 36.2%

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

   1. *-commutative 36.2%

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

8. Simplified 36.2%

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

9. Taylor expanded in l around 0 12.5%

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

10. Final simplification 12.5%

    \[\leadsto \cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right) \]
11. Add Preprocessing

Alternative 10: 9.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos (- (/ (* K m) 2.0) M)) (+ 1.0 (* l (+ (* l 0.5) -1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
}

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((((k * m) / 2.0d0) - m_1)) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * m) / 2.0) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * m) / 2.0) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 36.2%

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

   1. mul-1-neg 36.2%

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

5. Simplified 36.2%

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

6. Taylor expanded in m around inf 36.2%

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

   1. *-commutative 36.2%

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

8. Simplified 36.2%

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

9. Taylor expanded in l around 0 11.8%

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

10. Final simplification 11.8%

    \[\leadsto \cos \left(\frac{K \cdot m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \]
11. Add Preprocessing

Alternative 11: 7.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (cos M))

C

double code(double K, double m, double n, double M, double l) {
	return cos(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 = cos(m_1)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M);
}

Python

def code(K, m, n, M, l):
	return math.cos(M)

Julia

function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

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 36.2%

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

   1. mul-1-neg 36.2%

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

5. Simplified 36.2%

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

6. Taylor expanded in l around 0 7.7%

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

7. Taylor expanded in K around 0 8.0%

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

   1. cos-neg 8.0%

      \[\leadsto \color{blue}{\cos M} \]

9. Simplified 8.0%

   \[\leadsto \color{blue}{\cos M} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))))))