Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.3%

Time: 28.9s

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: 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: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ t_1 := e^{t\_0 - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_2 := \left(m + n\right) \cdot 0.5\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_1 \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{\frac{K \cdot \left({m}^{3} + {n}^{3}\right)}{\mathsf{fma}\left(m, m, n \cdot \left(n - m\right)\right)}}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(t\_2 - M\right) \cdot \left(M - t\_2\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- m n)) l))
        (t_1 (exp (- t_0 (pow (- (/ (+ m n) 2.0) M) 2.0))))
        (t_2 (* (+ m n) 0.5)))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_1) -1e-24)
     (*
      t_1
      (cos
       (-
        (/ (/ (* K (+ (pow m 3.0) (pow n 3.0))) (fma m m (* n (- n m)))) 2.0)
        M)))
     (exp (+ (* (- t_2 M) (- M t_2)) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n)) - l;
	double t_1 = exp((t_0 - pow((((m + n) / 2.0) - M), 2.0)));
	double t_2 = (m + n) * 0.5;
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_1) <= -1e-24) {
		tmp = t_1 * cos(((((K * (pow(m, 3.0) + pow(n, 3.0))) / fma(m, m, (n * (n - m)))) / 2.0) - M));
	} else {
		tmp = exp((((t_2 - M) * (M - t_2)) + t_0));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	t_1 = exp(Float64(t_0 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	t_2 = Float64(Float64(m + n) * 0.5)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_1) <= -1e-24)
		tmp = Float64(t_1 * cos(Float64(Float64(Float64(Float64(K * Float64((m ^ 3.0) + (n ^ 3.0))) / fma(m, m, Float64(n * Float64(n - m)))) / 2.0) - M)));
	else
		tmp = exp(Float64(Float64(Float64(t_2 - M) * Float64(M - t_2)) + t_0));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(t$95$0 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$1), $MachinePrecision], -1e-24], N[(t$95$1 * N[Cos[N[(N[(N[(N[(K * N[(N[Power[m, 3.0], $MachinePrecision] + N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(m * m + N[(n * N[(n - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[(t$95$2 - M), $MachinePrecision] * N[(M - t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$0), $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)))))) < -9.99999999999999924e-25

   1. Initial program 59.5%

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

      1. *-commutative 59.5%

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

      2. flip3-+ 59.6%

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

      3. associate-*l/ 74.0%

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

      4. fma-define 74.0%

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

      5. distribute-rgt-out-- 74.0%

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

   4. Applied egg-rr 74.0%

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

   if -9.99999999999999924e-25 < (*.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 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. unpow2 77.8%

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

      2. div-inv 77.8%

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

      3. metadata-eval 77.8%

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

      4. div-inv 77.8%

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

      5. metadata-eval 77.8%

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

   4. Applied egg-rr 77.8%

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

   5. Taylor expanded in m around 0 86.3%

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

   6. Taylor expanded in M around 0 86.0%

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

   7. Taylor expanded in K around 0 98.6%

      \[\leadsto \color{blue}{1} \cdot e^{\left(-\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{\frac{K \cdot \left({m}^{3} + {n}^{3}\right)}{\mathsf{fma}\left(m, m, n \cdot \left(n - m\right)\right)}}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(M - \left(m + n\right) \cdot 0.5\right) + \left(\left|m - n\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5\\ t_1 := \left|m - n\right| - \ell\\ t_2 := e^{\left(t\_0 - M\right) \cdot \left(M - t\_0\right) + t\_1}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{t\_1 - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq 1:\\ \;\;\;\;t\_2 \cdot \cos \left(\frac{K \cdot n}{2} - M\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (+ m n) 0.5))
        (t_1 (- (fabs (- m n)) l))
        (t_2 (exp (+ (* (- t_0 M) (- M t_0)) t_1))))
   (if (<=
        (*
         (cos (- (/ (* K (+ m n)) 2.0) M))
         (exp (- t_1 (pow (- (/ (+ m n) 2.0) M) 2.0))))
        1.0)
     (* t_2 (cos (- (/ (* K n) 2.0) M)))
     t_2)))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * 0.5;
	double t_1 = fabs((m - n)) - l;
	double t_2 = exp((((t_0 - M) * (M - t_0)) + t_1));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * exp((t_1 - pow((((m + n) / 2.0) - M), 2.0)))) <= 1.0) {
		tmp = t_2 * cos((((K * n) / 2.0) - M));
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (m + n) * 0.5d0
    t_1 = abs((m - n)) - l
    t_2 = exp((((t_0 - m_1) * (m_1 - t_0)) + t_1))
    if ((cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((t_1 - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))) <= 1.0d0) then
        tmp = t_2 * cos((((k * n) / 2.0d0) - m_1))
    else
        tmp = t_2
    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 t_1 = Math.abs((m - n)) - l;
	double t_2 = Math.exp((((t_0 - M) * (M - t_0)) + t_1));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((t_1 - Math.pow((((m + n) / 2.0) - M), 2.0)))) <= 1.0) {
		tmp = t_2 * Math.cos((((K * n) / 2.0) - M));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m + n) * 0.5
	t_1 = math.fabs((m - n)) - l
	t_2 = math.exp((((t_0 - M) * (M - t_0)) + t_1))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((t_1 - math.pow((((m + n) / 2.0) - M), 2.0)))) <= 1.0:
		tmp = t_2 * math.cos((((K * n) / 2.0) - M))
	else:
		tmp = t_2
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * 0.5)
	t_1 = Float64(abs(Float64(m - n)) - l)
	t_2 = exp(Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) + t_1))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(t_1 - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))) <= 1.0)
		tmp = Float64(t_2 * cos(Float64(Float64(Float64(K * n) / 2.0) - M)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m + n) * 0.5;
	t_1 = abs((m - n)) - l;
	t_2 = exp((((t_0 - M) * (M - t_0)) + t_1));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * exp((t_1 - ((((m + n) / 2.0) - M) ^ 2.0)))) <= 1.0)
		tmp = t_2 * cos((((K * n) / 2.0) - M));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $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[(t$95$1 - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0], N[(t$95$2 * N[Cos[N[(N[(N[(K * n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

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)))))) < 1

   1. Initial program 96.7%

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

      1. unpow2 96.7%

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

      2. div-inv 96.7%

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

      3. metadata-eval 96.7%

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

      4. div-inv 96.7%

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

      5. metadata-eval 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in m around 0 96.8%

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

   if 1 < (*.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 16.1%

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

      1. unpow2 16.1%

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

      2. div-inv 16.1%

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

      3. metadata-eval 16.1%

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

      4. div-inv 16.1%

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

      5. metadata-eval 16.1%

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

   4. Applied egg-rr 16.1%

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

   5. Taylor expanded in m around 0 50.0%

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

   6. Taylor expanded in M around 0 50.0%

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

   7. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{1} \cdot e^{\left(-\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

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

Alternative 3: 45.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(M - n\right) - \left(n + \ell\right)}\\ \mathbf{if}\;M \leq -58000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq -1.1 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (* M (- M n)) (+ n l))))))
   (if (<= M -58000000000.0)
     (* (cos M) (exp (* n M)))
     (if (<= M -1.1e-90)
       t_0
       (if (<= M -3.8e-140)
         (* (cos M) (exp (* n (- M (* m 0.5)))))
         (if (<= M 2.7e+82)
           t_0
           (*
            (cos (- (* K (* m 0.5)) M))
            (exp (* n (+ (* m 0.5) (- -1.0 M)))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(((M * (M - n)) - (n + l)));
	double tmp;
	if (M <= -58000000000.0) {
		tmp = cos(M) * exp((n * M));
	} else if (M <= -1.1e-90) {
		tmp = t_0;
	} else if (M <= -3.8e-140) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else if (M <= 2.7e+82) {
		tmp = t_0;
	} else {
		tmp = cos(((K * (m * 0.5)) - M)) * exp((n * ((m * 0.5) + (-1.0 - 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 = cos(m_1) * exp(((m_1 * (m_1 - n)) - (n + l)))
    if (m_1 <= (-58000000000.0d0)) then
        tmp = cos(m_1) * exp((n * m_1))
    else if (m_1 <= (-1.1d-90)) then
        tmp = t_0
    else if (m_1 <= (-3.8d-140)) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else if (m_1 <= 2.7d+82) then
        tmp = t_0
    else
        tmp = cos(((k * (m * 0.5d0)) - m_1)) * exp((n * ((m * 0.5d0) + ((-1.0d0) - m_1))))
    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(M) * Math.exp(((M * (M - n)) - (n + l)));
	double tmp;
	if (M <= -58000000000.0) {
		tmp = Math.cos(M) * Math.exp((n * M));
	} else if (M <= -1.1e-90) {
		tmp = t_0;
	} else if (M <= -3.8e-140) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else if (M <= 2.7e+82) {
		tmp = t_0;
	} else {
		tmp = Math.cos(((K * (m * 0.5)) - M)) * Math.exp((n * ((m * 0.5) + (-1.0 - M))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(((M * (M - n)) - (n + l)))
	tmp = 0
	if M <= -58000000000.0:
		tmp = math.cos(M) * math.exp((n * M))
	elif M <= -1.1e-90:
		tmp = t_0
	elif M <= -3.8e-140:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	elif M <= 2.7e+82:
		tmp = t_0
	else:
		tmp = math.cos(((K * (m * 0.5)) - M)) * math.exp((n * ((m * 0.5) + (-1.0 - M))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(Float64(M * Float64(M - n)) - Float64(n + l))))
	tmp = 0.0
	if (M <= -58000000000.0)
		tmp = Float64(cos(M) * exp(Float64(n * M)));
	elseif (M <= -1.1e-90)
		tmp = t_0;
	elseif (M <= -3.8e-140)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	elseif (M <= 2.7e+82)
		tmp = t_0;
	else
		tmp = Float64(cos(Float64(Float64(K * Float64(m * 0.5)) - M)) * exp(Float64(n * Float64(Float64(m * 0.5) + Float64(-1.0 - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(((M * (M - n)) - (n + l)));
	tmp = 0.0;
	if (M <= -58000000000.0)
		tmp = cos(M) * exp((n * M));
	elseif (M <= -1.1e-90)
		tmp = t_0;
	elseif (M <= -3.8e-140)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	elseif (M <= 2.7e+82)
		tmp = t_0;
	else
		tmp = cos(((K * (m * 0.5)) - M)) * exp((n * ((m * 0.5) + (-1.0 - M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(M - n), $MachinePrecision]), $MachinePrecision] - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -58000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -1.1e-90], t$95$0, If[LessEqual[M, -3.8e-140], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.7e+82], t$95$0, N[(N[Cos[N[(N[(K * N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(n * N[(N[(m * 0.5), $MachinePrecision] + N[(-1.0 - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -5.8e10

   1. Initial program 76.7%

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

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in n around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. distribute-rgt-out 93.4%

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

      4. *-commutative 93.4%

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

      5. *-commutative 93.4%

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

   8. Simplified 93.4%

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

   9. Taylor expanded in n around inf 42.9%

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

   10. Taylor expanded in M around inf 44.6%

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

       1. *-commutative 44.6%

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

   12. Simplified 44.6%

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

   if -5.8e10 < M < -1.09999999999999993e-90 or -3.79999999999999998e-140 < M < 2.6999999999999999e82

   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. Step-by-step derivation

      1. *-un-lft-identity 78.1%

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

      2. *-commutative 78.1%

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

   4. Applied egg-rr 25.5%

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

   5. Taylor expanded in n around 0 39.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 39.6%

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

      2. unpow2 39.6%

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

      3. distribute-rgt-out 40.3%

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

   7. Simplified 40.3%

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

   8. Taylor expanded in n around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-*l* 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in m around 0 53.4%

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

       1. cos-neg 53.4%

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

       2. associate-*r* 53.4%

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

       3. neg-mul-1 53.4%

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

       4. +-commutative 53.4%

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

   13. Simplified 53.4%

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

   if -1.09999999999999993e-90 < M < -3.79999999999999998e-140

   1. Initial program 60.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. distribute-rgt-out 70.5%

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

      4. *-commutative 70.5%

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

      5. *-commutative 70.5%

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

   8. Simplified 70.5%

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

   9. Taylor expanded in n around inf 70.6%

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

   if 2.6999999999999999e82 < M

   1. Initial program 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. *-commutative 78.7%

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

   4. Applied egg-rr 3.3%

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

   5. Taylor expanded in n around 0 3.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 3.2%

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

      2. unpow2 3.2%

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

      3. distribute-rgt-out 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in n around 0 9.7%

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

      1. *-commutative 9.7%

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

      2. associate-*l* 9.7%

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

   10. Simplified 9.7%

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

   11. Taylor expanded in n around inf 28.8%

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

       1. +-commutative 28.8%

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

   13. Simplified 28.8%

       \[\leadsto \cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot \left(e^{\color{blue}{n \cdot \left(0.5 \cdot m - \left(M + 1\right)\right)}} \cdot 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -58000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq -1.1 \cdot 10^{-90}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) - \left(n + \ell\right)}\\ \mathbf{elif}\;M \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -6000000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq -9.8 \cdot 10^{-91} \lor \neg \left(M \leq -3.8 \cdot 10^{-140}\right) \land M \leq 3 \cdot 10^{+87}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -6000000000000.0)
   (* (cos M) (exp (* n M)))
   (if (or (<= M -9.8e-91) (and (not (<= M -3.8e-140)) (<= M 3e+87)))
     (* (cos M) (exp (- (* M (- M n)) (+ n l))))
     (* (cos M) (exp (* n (- M (* m 0.5))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -6000000000000.0) {
		tmp = cos(M) * exp((n * M));
	} else if ((M <= -9.8e-91) || (!(M <= -3.8e-140) && (M <= 3e+87))) {
		tmp = cos(M) * exp(((M * (M - n)) - (n + l)));
	} else {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	}
	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 (m_1 <= (-6000000000000.0d0)) then
        tmp = cos(m_1) * exp((n * m_1))
    else if ((m_1 <= (-9.8d-91)) .or. (.not. (m_1 <= (-3.8d-140))) .and. (m_1 <= 3d+87)) then
        tmp = cos(m_1) * exp(((m_1 * (m_1 - n)) - (n + l)))
    else
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -6000000000000.0) {
		tmp = Math.cos(M) * Math.exp((n * M));
	} else if ((M <= -9.8e-91) || (!(M <= -3.8e-140) && (M <= 3e+87))) {
		tmp = Math.cos(M) * Math.exp(((M * (M - n)) - (n + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -6000000000000.0:
		tmp = math.cos(M) * math.exp((n * M))
	elif (M <= -9.8e-91) or (not (M <= -3.8e-140) and (M <= 3e+87)):
		tmp = math.cos(M) * math.exp(((M * (M - n)) - (n + l)))
	else:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -6000000000000.0)
		tmp = Float64(cos(M) * exp(Float64(n * M)));
	elseif ((M <= -9.8e-91) || (!(M <= -3.8e-140) && (M <= 3e+87)))
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(M - n)) - Float64(n + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -6000000000000.0)
		tmp = cos(M) * exp((n * M));
	elseif ((M <= -9.8e-91) || (~((M <= -3.8e-140)) && (M <= 3e+87)))
		tmp = cos(M) * exp(((M * (M - n)) - (n + l)));
	else
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -6000000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[M, -9.8e-91], And[N[Not[LessEqual[M, -3.8e-140]], $MachinePrecision], LessEqual[M, 3e+87]]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(M - n), $MachinePrecision]), $MachinePrecision] - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -6e12

   1. Initial program 76.7%

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

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in n around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. distribute-rgt-out 93.4%

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

      4. *-commutative 93.4%

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

      5. *-commutative 93.4%

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

   8. Simplified 93.4%

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

   9. Taylor expanded in n around inf 42.9%

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

   10. Taylor expanded in M around inf 44.6%

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

       1. *-commutative 44.6%

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

   12. Simplified 44.6%

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

   if -6e12 < M < -9.7999999999999996e-91 or -3.79999999999999998e-140 < M < 2.9999999999999999e87

   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. Step-by-step derivation

      1. *-un-lft-identity 78.1%

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

      2. *-commutative 78.1%

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

   4. Applied egg-rr 25.5%

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

   5. Taylor expanded in n around 0 39.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 39.6%

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

      2. unpow2 39.6%

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

      3. distribute-rgt-out 40.3%

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

   7. Simplified 40.3%

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

   8. Taylor expanded in n around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-*l* 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in m around 0 53.4%

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

       1. cos-neg 53.4%

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

       2. associate-*r* 53.4%

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

       3. neg-mul-1 53.4%

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

       4. +-commutative 53.4%

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

   13. Simplified 53.4%

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

   if -9.7999999999999996e-91 < M < -3.79999999999999998e-140 or 2.9999999999999999e87 < M

   1. Initial program 75.4%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. unpow2 82.5%

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

      3. distribute-rgt-out 89.6%

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

      4. *-commutative 89.6%

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

      5. *-commutative 89.6%

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

   8. Simplified 89.6%

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

   9. Taylor expanded in n around inf 55.3%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6000000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq -9.8 \cdot 10^{-91} \lor \neg \left(M \leq -3.8 \cdot 10^{-140}\right) \land M \leq 3 \cdot 10^{+87}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot \left(m \cdot 0.5\right) - M\right)\\ \mathbf{if}\;M \leq -27000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq 10^{-58}:\\ \;\;\;\;t\_0 \cdot e^{\left(m - n\right) - \left(\ell + M \cdot \left(n - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot e^{\left(m - n\right) + \left(n \cdot \left(m \cdot 0.5 - M\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (* K (* m 0.5)) M))))
   (if (<= M -27000000000.0)
     (* (cos M) (exp (* n M)))
     (if (<= M 1e-58)
       (* t_0 (exp (- (- m n) (+ l (* M (- n M))))))
       (* t_0 (exp (+ (- m n) (- (* n (- (* m 0.5) M)) l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((K * (m * 0.5)) - M));
	double tmp;
	if (M <= -27000000000.0) {
		tmp = cos(M) * exp((n * M));
	} else if (M <= 1e-58) {
		tmp = t_0 * exp(((m - n) - (l + (M * (n - M)))));
	} else {
		tmp = t_0 * exp(((m - n) + ((n * ((m * 0.5) - M)) - 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 * 0.5d0)) - m_1))
    if (m_1 <= (-27000000000.0d0)) then
        tmp = cos(m_1) * exp((n * m_1))
    else if (m_1 <= 1d-58) then
        tmp = t_0 * exp(((m - n) - (l + (m_1 * (n - m_1)))))
    else
        tmp = t_0 * exp(((m - n) + ((n * ((m * 0.5d0) - m_1)) - 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 * 0.5)) - M));
	double tmp;
	if (M <= -27000000000.0) {
		tmp = Math.cos(M) * Math.exp((n * M));
	} else if (M <= 1e-58) {
		tmp = t_0 * Math.exp(((m - n) - (l + (M * (n - M)))));
	} else {
		tmp = t_0 * Math.exp(((m - n) + ((n * ((m * 0.5) - M)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((K * (m * 0.5)) - M))
	tmp = 0
	if M <= -27000000000.0:
		tmp = math.cos(M) * math.exp((n * M))
	elif M <= 1e-58:
		tmp = t_0 * math.exp(((m - n) - (l + (M * (n - M)))))
	else:
		tmp = t_0 * math.exp(((m - n) + ((n * ((m * 0.5) - M)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(K * Float64(m * 0.5)) - M))
	tmp = 0.0
	if (M <= -27000000000.0)
		tmp = Float64(cos(M) * exp(Float64(n * M)));
	elseif (M <= 1e-58)
		tmp = Float64(t_0 * exp(Float64(Float64(m - n) - Float64(l + Float64(M * Float64(n - M))))));
	else
		tmp = Float64(t_0 * exp(Float64(Float64(m - n) + Float64(Float64(n * Float64(Float64(m * 0.5) - M)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((K * (m * 0.5)) - M));
	tmp = 0.0;
	if (M <= -27000000000.0)
		tmp = cos(M) * exp((n * M));
	elseif (M <= 1e-58)
		tmp = t_0 * exp(((m - n) - (l + (M * (n - M)))));
	else
		tmp = t_0 * exp(((m - n) + ((n * ((m * 0.5) - M)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(K * N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -27000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1e-58], N[(t$95$0 * N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(n * N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.7e10

   1. Initial program 76.7%

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

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in n around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. distribute-rgt-out 93.4%

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

      4. *-commutative 93.4%

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

      5. *-commutative 93.4%

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

   8. Simplified 93.4%

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

   9. Taylor expanded in n around inf 42.9%

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

   10. Taylor expanded in M around inf 44.6%

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

       1. *-commutative 44.6%

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

   12. Simplified 44.6%

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

   if -2.7e10 < M < 1e-58

   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. Step-by-step derivation

      1. *-un-lft-identity 77.3%

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

      2. *-commutative 77.3%

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

   4. Applied egg-rr 24.2%

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

   5. Taylor expanded in n around 0 37.2%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 37.2%

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

      2. unpow2 37.2%

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

      3. distribute-rgt-out 38.0%

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

   7. Simplified 38.0%

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

   8. Taylor expanded in n around 0 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*l* 38.8%

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

   10. Simplified 38.8%

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

   11. Taylor expanded in m around 0 54.3%

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

       1. associate-*r* 54.3%

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

       2. neg-mul-1 54.3%

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

   13. Simplified 54.3%

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

   if 1e-58 < M

   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. Step-by-step derivation

      1. *-un-lft-identity 77.3%

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

      2. *-commutative 77.3%

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

   4. Applied egg-rr 10.4%

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

   5. Taylor expanded in n around 0 16.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 16.8%

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

      2. unpow2 16.8%

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

      3. distribute-rgt-out 19.6%

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

   7. Simplified 19.6%

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

   8. Taylor expanded in n around 0 23.6%

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

      1. *-commutative 23.6%

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

      2. associate-*l* 23.6%

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

   10. Simplified 23.6%

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

   11. Taylor expanded in n around inf 43.5%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq 10^{-58}:\\ \;\;\;\;\cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot e^{\left(m - n\right) - \left(\ell + M \cdot \left(n - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot e^{\left(m - n\right) + \left(n \cdot \left(m \cdot 0.5 - M\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot \left(m \cdot 0.5\right) - M\right)\\ \mathbf{if}\;M \leq -31000000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{+110}:\\ \;\;\;\;t\_0 \cdot e^{\left(m - n\right) - \left(\ell + M \cdot \left(n - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (* K (* m 0.5)) M))))
   (if (<= M -31000000000000.0)
     (* (cos M) (exp (* n M)))
     (if (<= M 1.35e+110)
       (* t_0 (exp (- (- m n) (+ l (* M (- n M))))))
       (* t_0 (exp (* n (+ (* m 0.5) (- -1.0 M)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((K * (m * 0.5)) - M));
	double tmp;
	if (M <= -31000000000000.0) {
		tmp = cos(M) * exp((n * M));
	} else if (M <= 1.35e+110) {
		tmp = t_0 * exp(((m - n) - (l + (M * (n - M)))));
	} else {
		tmp = t_0 * exp((n * ((m * 0.5) + (-1.0 - 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 = cos(((k * (m * 0.5d0)) - m_1))
    if (m_1 <= (-31000000000000.0d0)) then
        tmp = cos(m_1) * exp((n * m_1))
    else if (m_1 <= 1.35d+110) then
        tmp = t_0 * exp(((m - n) - (l + (m_1 * (n - m_1)))))
    else
        tmp = t_0 * exp((n * ((m * 0.5d0) + ((-1.0d0) - m_1))))
    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 * 0.5)) - M));
	double tmp;
	if (M <= -31000000000000.0) {
		tmp = Math.cos(M) * Math.exp((n * M));
	} else if (M <= 1.35e+110) {
		tmp = t_0 * Math.exp(((m - n) - (l + (M * (n - M)))));
	} else {
		tmp = t_0 * Math.exp((n * ((m * 0.5) + (-1.0 - M))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((K * (m * 0.5)) - M))
	tmp = 0
	if M <= -31000000000000.0:
		tmp = math.cos(M) * math.exp((n * M))
	elif M <= 1.35e+110:
		tmp = t_0 * math.exp(((m - n) - (l + (M * (n - M)))))
	else:
		tmp = t_0 * math.exp((n * ((m * 0.5) + (-1.0 - M))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(K * Float64(m * 0.5)) - M))
	tmp = 0.0
	if (M <= -31000000000000.0)
		tmp = Float64(cos(M) * exp(Float64(n * M)));
	elseif (M <= 1.35e+110)
		tmp = Float64(t_0 * exp(Float64(Float64(m - n) - Float64(l + Float64(M * Float64(n - M))))));
	else
		tmp = Float64(t_0 * exp(Float64(n * Float64(Float64(m * 0.5) + Float64(-1.0 - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((K * (m * 0.5)) - M));
	tmp = 0.0;
	if (M <= -31000000000000.0)
		tmp = cos(M) * exp((n * M));
	elseif (M <= 1.35e+110)
		tmp = t_0 * exp(((m - n) - (l + (M * (n - M)))));
	else
		tmp = t_0 * exp((n * ((m * 0.5) + (-1.0 - M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(K * N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -31000000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.35e+110], N[(t$95$0 * N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(n * N[(N[(m * 0.5), $MachinePrecision] + N[(-1.0 - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -3.1e13

   1. Initial program 76.7%

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

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in n around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. distribute-rgt-out 93.4%

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

      4. *-commutative 93.4%

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

      5. *-commutative 93.4%

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

   8. Simplified 93.4%

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

   9. Taylor expanded in n around inf 42.9%

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

   10. Taylor expanded in M around inf 44.6%

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

       1. *-commutative 44.6%

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

   12. Simplified 44.6%

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

   if -3.1e13 < M < 1.35000000000000005e110

   1. Initial program 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. *-commutative 77.0%

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

   4. Applied egg-rr 23.7%

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

   5. Taylor expanded in n around 0 37.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 37.1%

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

      2. unpow2 37.1%

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

      3. distribute-rgt-out 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in n around 0 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-*l* 39.7%

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

   10. Simplified 39.7%

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

   11. Taylor expanded in m around 0 53.1%

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

       1. associate-*r* 53.1%

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

       2. neg-mul-1 53.1%

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

   13. Simplified 53.1%

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

   if 1.35000000000000005e110 < M

   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. Step-by-step derivation

      1. *-un-lft-identity 78.6%

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

      2. *-commutative 78.6%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in n around 0 1.0%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \left(e^{\left(m - n\right) + \left(\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)} - \ell\right)} \cdot 1\right) \]
   6. Step-by-step derivation

      1. +-commutative 1.0%

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

      2. unpow2 1.0%

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

      3. distribute-rgt-out 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in n around 0 8.4%

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

      1. *-commutative 8.4%

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

      2. associate-*l* 8.4%

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

   10. Simplified 8.4%

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

   11. Taylor expanded in n around inf 32.0%

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

       1. +-commutative 32.0%

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

   13. Simplified 32.0%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -31000000000000:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{+110}:\\ \;\;\;\;\cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot e^{\left(m - n\right) - \left(\ell + M \cdot \left(n - M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot \left(m \cdot 0.5\right) - M\right) \cdot e^{n \cdot \left(m \cdot 0.5 + \left(-1 - M\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5\\ e^{\left(t\_0 - M\right) \cdot \left(M - t\_0\right) + \left(\left|m - n\right| - \ell\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (+ m n) 0.5)))
   (exp (+ (* (- t_0 M) (- M t_0)) (- (fabs (- m n)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * 0.5;
	return exp((((t_0 - M) * (M - t_0)) + (fabs((m - n)) - 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
    real(8) :: t_0
    t_0 = (m + n) * 0.5d0
    code = exp((((t_0 - m_1) * (m_1 - t_0)) + (abs((m - n)) - l)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * 0.5;
	return Math.exp((((t_0 - M) * (M - t_0)) + (Math.abs((m - n)) - l)));
}

Python

def code(K, m, n, M, l):
	t_0 = (m + n) * 0.5
	return math.exp((((t_0 - M) * (M - t_0)) + (math.fabs((m - n)) - l)))

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * 0.5)
	return exp(Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) + Float64(abs(Float64(m - n)) - l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	t_0 = (m + n) * 0.5;
	tmp = exp((((t_0 - M) * (M - t_0)) + (abs((m - n)) - l)));
end

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. unpow2 77.2%

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

   2. div-inv 77.2%

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

   3. metadata-eval 77.2%

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

   4. div-inv 77.2%

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

   5. metadata-eval 77.2%

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

4. Applied egg-rr 77.2%

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

5. Taylor expanded in m around 0 85.4%

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

6. Taylor expanded in M around 0 84.8%

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

7. Taylor expanded in K around 0 95.6%

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

8. Final simplification 95.6%

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

Alternative 8: 36.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -7.5e-84)
   (* (cos M) (exp (* n M)))
   (* (cos M) (exp (* n (- M (* m 0.5)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -7.5e-84) {
		tmp = cos(M) * exp((n * M));
	} else {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	}
	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 (m_1 <= (-7.5d-84)) then
        tmp = cos(m_1) * exp((n * m_1))
    else
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -7.5e-84) {
		tmp = Math.cos(M) * Math.exp((n * M));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -7.5e-84:
		tmp = math.cos(M) * math.exp((n * M))
	else:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -7.5e-84)
		tmp = Float64(cos(M) * exp(Float64(n * M)));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -7.5e-84)
		tmp = cos(M) * exp((n * M));
	else
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -7.5e-84], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -7.50000000000000026e-84

   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 K around 0 98.8%

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

      1. cos-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in n around 0 71.3%

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

      1. +-commutative 71.3%

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

      2. unpow2 71.3%

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

      3. distribute-rgt-out 84.6%

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

      4. *-commutative 84.6%

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

      5. *-commutative 84.6%

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

   8. Simplified 84.6%

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

   9. Taylor expanded in n around inf 37.8%

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

   10. Taylor expanded in M around inf 41.4%

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

       1. *-commutative 41.4%

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

   12. Simplified 41.4%

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

   if -7.50000000000000026e-84 < M

   1. Initial program 76.0%

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

   3. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in n around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 77.5%

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

      4. *-commutative 77.5%

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

      5. *-commutative 77.5%

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

   8. Simplified 77.5%

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

   9. Taylor expanded in n around inf 40.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -3.9 \cdot 10^{-92}:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -3.9e-92)
   (* (cos M) (exp (* n M)))
   (* (cos M) (exp (* n (* m -0.5))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -3.9e-92) {
		tmp = cos(M) * exp((n * M));
	} else {
		tmp = cos(M) * exp((n * (m * -0.5)));
	}
	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 (m_1 <= (-3.9d-92)) then
        tmp = cos(m_1) * exp((n * m_1))
    else
        tmp = cos(m_1) * exp((n * (m * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -3.9e-92) {
		tmp = Math.cos(M) * Math.exp((n * M));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (m * -0.5)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -3.9e-92:
		tmp = math.cos(M) * math.exp((n * M))
	else:
		tmp = math.cos(M) * math.exp((n * (m * -0.5)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -3.9e-92)
		tmp = Float64(cos(M) * exp(Float64(n * M)));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(m * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -3.9e-92)
		tmp = cos(M) * exp((n * M));
	else
		tmp = cos(M) * exp((n * (m * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -3.9e-92], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(m * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.8999999999999997e-92

   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 K around 0 98.8%

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

      1. cos-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in n around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. distribute-rgt-out 83.6%

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

      4. *-commutative 83.6%

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

      5. *-commutative 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in n around inf 38.6%

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

   10. Taylor expanded in M around inf 42.1%

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

       1. *-commutative 42.1%

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

   12. Simplified 42.1%

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

   if -3.8999999999999997e-92 < M

   1. Initial program 76.5%

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

   3. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in n around 0 72.6%

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

      1. +-commutative 72.6%

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

      2. unpow2 72.6%

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

      3. distribute-rgt-out 77.9%

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

      4. *-commutative 77.9%

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

      5. *-commutative 77.9%

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

   8. Simplified 77.9%

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

   9. Taylor expanded in n around inf 40.1%

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

   10. Taylor expanded in M around 0 31.7%

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

       1. associate-*r* 31.7%

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

       2. *-commutative 31.7%

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

   12. Simplified 31.7%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.9 \cdot 10^{-92}:\\ \;\;\;\;\cos M \cdot e^{n \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{n \cdot M} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(n * M)))
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in K around 0 95.9%

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

   1. cos-neg 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in n around 0 71.9%

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

   1. +-commutative 71.9%

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

   2. unpow2 71.9%

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

   3. distribute-rgt-out 79.8%

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

   4. *-commutative 79.8%

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

   5. *-commutative 79.8%

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

8. Simplified 79.8%

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

9. Taylor expanded in n around inf 39.6%

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

10. Taylor expanded in M around inf 33.7%

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

    1. *-commutative 33.7%

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

12. Simplified 33.7%

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

13. Final simplification 33.7%

    \[\leadsto \cos M \cdot e^{n \cdot M} \]
14. Add Preprocessing

Alternative 11: 6.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * (1.0 + (n * (M - (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 = cos(m_1) * (1.0d0 + (n * (m_1 - (m * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in K around 0 95.9%

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

   1. cos-neg 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in n around 0 71.9%

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

   1. +-commutative 71.9%

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

   2. unpow2 71.9%

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

   3. distribute-rgt-out 79.8%

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

   4. *-commutative 79.8%

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

   5. *-commutative 79.8%

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

8. Simplified 79.8%

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

9. Taylor expanded in n around inf 39.6%

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

10. Taylor expanded in n around 0 6.1%

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

11. Final simplification 6.1%

    \[\leadsto \cos M \cdot \left(1 + n \cdot \left(M - m \cdot 0.5\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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)))))))