Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.4%

Time: 21.9s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. K = 6.339701882716489e+221
  2. m = -1.718782405678061e-145
  3. n = -1.2086589680395424e-263
  4. M = -1.2622660993941753e-264
  5. l = 3.073010118965153e-110

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: 76.4% 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Final simplification 97.1%

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

Alternative 2: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.15 \cdot 10^{+76} \lor \neg \left(M \leq 5 \cdot 10^{+58}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.15e+76) (not (<= M 5e+58)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (- m (+ n l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.15e+76) || !(M <= 5e+58)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((m - (n + l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1.15d+76)) .or. (.not. (m_1 <= 5d+58))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp((m - (n + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.15e+76) || !(M <= 5e+58)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((m - (n + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.15e+76) or not (M <= 5e+58):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((m - (n + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.15e+76) || !(M <= 5e+58))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(m - Float64(n + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.15e+76) || ~((M <= 5e+58)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((m - (n + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.15e+76], N[Not[LessEqual[M, 5e+58]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.15000000000000001e76 or 4.99999999999999986e58 < M

   1. Initial program 73.1%

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

   3. Taylor expanded in 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 M around inf 97.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.9%

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

   8. Simplified 97.9%

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

   if -1.15000000000000001e76 < M < 4.99999999999999986e58

   1. Initial program 73.1%

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

   3. Taylor expanded in M around inf 37.5%

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

      1. associate-*r* 37.5%

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

      2. neg-mul-1 37.5%

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

      3. unpow2 37.5%

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

      4. sqr-neg 37.5%

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

      5. distribute-lft-out 37.5%

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

      6. unsub-neg 37.5%

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

   5. Simplified 37.5%

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

      1. sub-neg 37.5%

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

      2. add-sqr-sqrt 17.8%

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

      3. fabs-sqr 17.8%

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

      4. add-sqr-sqrt 53.6%

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

   7. Applied egg-rr 53.6%

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

      1. sub-neg 53.6%

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

      2. associate--r- 53.6%

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

   9. Simplified 53.6%

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

   10. Taylor expanded in K around 0 68.1%

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

       1. cos-neg 95.5%

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

   12. Simplified 68.1%

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

   13. Taylor expanded in M around 0 68.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.15 \cdot 10^{+76} \lor \neg \left(M \leq 5 \cdot 10^{+58}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{+22}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(\left(m + n\right) - M\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.5e+22)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (* (cos M) (exp (+ (* M (- (+ m n) M)) (- (- m l) n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.5e+22) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((M * ((m + n) - M)) + ((m - l) - n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-2.5d+22)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((m_1 * ((m + n) - m_1)) + ((m - l) - n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.5e+22) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * ((m + n) - M)) + ((m - l) - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.5e+22:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((M * ((m + n) - M)) + ((m - l) - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.5e+22)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(Float64(m + n) - M)) + Float64(Float64(m - l) - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.5e+22)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((M * ((m + n) - M)) + ((m - l) - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.5e+22], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(N[(m + n), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] + N[(N[(m - l), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.4999999999999998e22

   1. Initial program 53.1%

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

   3. Taylor expanded in 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 m around inf 98.5%

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

   if -2.4999999999999998e22 < m

   1. Initial program 79.8%

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

   3. Taylor expanded in M around inf 54.1%

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

      1. associate-*r* 54.1%

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

      2. neg-mul-1 54.1%

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

      3. unpow2 54.1%

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

      4. sqr-neg 54.1%

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

      5. distribute-lft-out 55.7%

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

      6. unsub-neg 55.7%

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

   5. Simplified 55.7%

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

      1. sub-neg 55.7%

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

      2. add-sqr-sqrt 29.7%

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

      3. fabs-sqr 29.7%

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

      4. add-sqr-sqrt 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. sub-neg 62.2%

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

      2. associate--r- 62.2%

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

   9. Simplified 62.2%

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

   10. Taylor expanded in K around 0 71.4%

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

       1. cos-neg 96.1%

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

   12. Simplified 71.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{+22}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(\left(m + n\right) - M\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(\left(m + n\right) - M\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1e+107)
   (* (cos M) (exp (- m (+ n l))))
   (* (cos M) (exp (+ (* M (- (+ m n) M)) (- (- m l) n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1e+107) {
		tmp = cos(M) * exp((m - (n + l)));
	} else {
		tmp = cos(M) * exp(((M * ((m + n) - M)) + ((m - l) - n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1d+107)) then
        tmp = cos(m_1) * exp((m - (n + l)))
    else
        tmp = cos(m_1) * exp(((m_1 * ((m + n) - m_1)) + ((m - l) - n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1e+107) {
		tmp = Math.cos(M) * Math.exp((m - (n + l)));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * ((m + n) - M)) + ((m - l) - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1e+107:
		tmp = math.cos(M) * math.exp((m - (n + l)))
	else:
		tmp = math.cos(M) * math.exp(((M * ((m + n) - M)) + ((m - l) - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1e+107)
		tmp = Float64(cos(M) * exp(Float64(m - Float64(n + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(Float64(m + n) - M)) + Float64(Float64(m - l) - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1e+107)
		tmp = cos(M) * exp((m - (n + l)));
	else
		tmp = cos(M) * exp(((M * ((m + n) - M)) + ((m - l) - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1e+107], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(N[(m + n), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] + N[(N[(m - l), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -9.9999999999999997e106

   1. Initial program 47.7%

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

   3. Taylor expanded in M around inf 18.6%

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

      1. associate-*r* 18.6%

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

      2. neg-mul-1 18.6%

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

      3. unpow2 18.6%

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

      4. sqr-neg 18.6%

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

      5. distribute-lft-out 20.9%

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

      6. unsub-neg 20.9%

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

   5. Simplified 20.9%

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

      1. sub-neg 20.9%

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

      2. add-sqr-sqrt 2.3%

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

      3. fabs-sqr 2.3%

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

      4. add-sqr-sqrt 43.3%

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

   7. Applied egg-rr 43.3%

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

      1. sub-neg 43.3%

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

      2. associate--r- 43.3%

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

   9. Simplified 43.3%

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

   10. Taylor expanded in K around 0 86.6%

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

       1. cos-neg 100.0%

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

   12. Simplified 86.6%

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

   13. Taylor expanded in M around 0 95.5%

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

   if -9.9999999999999997e106 < m

   1. Initial program 78.4%

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

   3. Taylor expanded in M around inf 51.4%

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

      1. associate-*r* 51.4%

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

      2. neg-mul-1 51.4%

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

      3. unpow2 51.4%

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

      4. sqr-neg 51.4%

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

      5. distribute-lft-out 53.3%

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

      6. unsub-neg 53.3%

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

   5. Simplified 53.3%

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

      1. sub-neg 53.3%

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

      2. add-sqr-sqrt 27.4%

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

      3. fabs-sqr 27.4%

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

      4. add-sqr-sqrt 61.1%

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

   7. Applied egg-rr 61.1%

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

      1. sub-neg 61.1%

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

      2. associate--r- 61.1%

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

   9. Simplified 61.1%

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

   10. Taylor expanded in K around 0 72.2%

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

       1. cos-neg 96.5%

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

   12. Simplified 72.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(\left(m + n\right) - M\right) + \left(\left(m - \ell\right) - n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m - \left(n + \ell\right)\\ \mathbf{if}\;m \leq -2 \cdot 10^{+106}:\\ \;\;\;\;\cos M \cdot e^{t\_0}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 + M \cdot \left(m + \left(n - M\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- m (+ n l))))
   (if (<= m -2e+106)
     (* (cos M) (exp t_0))
     (exp (+ t_0 (* M (+ m (- n M))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = m - (n + l);
	double tmp;
	if (m <= -2e+106) {
		tmp = cos(M) * exp(t_0);
	} else {
		tmp = exp((t_0 + (M * (m + (n - 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 = m - (n + l)
    if (m <= (-2d+106)) then
        tmp = cos(m_1) * exp(t_0)
    else
        tmp = exp((t_0 + (m_1 * (m + (n - 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 = m - (n + l);
	double tmp;
	if (m <= -2e+106) {
		tmp = Math.cos(M) * Math.exp(t_0);
	} else {
		tmp = Math.exp((t_0 + (M * (m + (n - M)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = m - (n + l)
	tmp = 0
	if m <= -2e+106:
		tmp = math.cos(M) * math.exp(t_0)
	else:
		tmp = math.exp((t_0 + (M * (m + (n - M)))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(m - Float64(n + l))
	tmp = 0.0
	if (m <= -2e+106)
		tmp = Float64(cos(M) * exp(t_0));
	else
		tmp = exp(Float64(t_0 + Float64(M * Float64(m + Float64(n - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = m - (n + l);
	tmp = 0.0;
	if (m <= -2e+106)
		tmp = cos(M) * exp(t_0);
	else
		tmp = exp((t_0 + (M * (m + (n - M)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2e+106], N[(N[Cos[M], $MachinePrecision] * N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 + N[(M * N[(m + N[(n - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.00000000000000018e106

   1. Initial program 46.7%

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

   3. Taylor expanded in M around inf 18.2%

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

      1. associate-*r* 18.2%

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

      2. neg-mul-1 18.2%

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

      3. unpow2 18.2%

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

      4. sqr-neg 18.2%

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

      5. distribute-lft-out 20.4%

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

      6. unsub-neg 20.4%

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

   5. Simplified 20.4%

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

      1. sub-neg 20.4%

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

      2. add-sqr-sqrt 2.2%

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

      3. fabs-sqr 2.2%

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

      4. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. sub-neg 42.3%

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

      2. associate--r- 42.3%

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

   9. Simplified 42.3%

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

   10. Taylor expanded in K around 0 86.9%

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

       1. cos-neg 100.0%

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

   12. Simplified 86.9%

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

   13. Taylor expanded in M around 0 93.4%

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

   if -2.00000000000000018e106 < m

   1. Initial program 78.8%

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

   3. Taylor expanded in M around inf 51.6%

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

      1. associate-*r* 51.6%

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

      2. neg-mul-1 51.6%

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

      3. unpow2 51.6%

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

      4. sqr-neg 51.6%

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

      5. distribute-lft-out 53.6%

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

      6. unsub-neg 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in m around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-*l* 56.3%

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

   8. Simplified 56.3%

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

   9. Taylor expanded in K around 0 59.6%

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

       1. mul-1-neg 59.6%

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

       2. associate-+r- 59.6%

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

       3. distribute-lft-neg-in 59.6%

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

       4. associate--r+ 59.6%

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

       5. cancel-sign-sub 59.6%

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

   11. Simplified 59.6%

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

       1. sub-neg 59.6%

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

       2. add-sqr-sqrt 30.8%

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

       3. fabs-sqr 30.8%

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

       4. add-sqr-sqrt 70.2%

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

   13. Applied egg-rr 70.2%

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

       1. sub-neg 70.2%

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

       2. associate--l- 70.2%

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

   15. Simplified 70.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+106}:\\ \;\;\;\;\cos M \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - \left(n + \ell\right)\right) + M \cdot \left(m + \left(n - M\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.000185:\\ \;\;\;\;e^{M \cdot \left(m + \left(n - M\right)\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.000185) (exp (- (* M (+ m (- n M))) l)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.000185) {
		tmp = exp(((M * (m + (n - M))) - l));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 0.000185d0) then
        tmp = exp(((m_1 * (m + (n - m_1))) - l))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.000185) {
		tmp = Math.exp(((M * (m + (n - M))) - l));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.000185:
		tmp = math.exp(((M * (m + (n - M))) - l))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.000185)
		tmp = exp(Float64(Float64(M * Float64(m + Float64(n - M))) - l));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.000185)
		tmp = exp(((M * (m + (n - M))) - l));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.000185], N[Exp[N[(N[(M * N[(m + N[(n - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.85e-4

   1. Initial program 73.0%

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

   3. Taylor expanded in M around inf 43.1%

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

      1. associate-*r* 43.1%

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

      2. neg-mul-1 43.1%

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

      3. unpow2 43.1%

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

      4. sqr-neg 43.1%

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

      5. distribute-lft-out 44.7%

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

      6. unsub-neg 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in m around inf 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-*l* 45.6%

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

   8. Simplified 45.6%

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

   9. Taylor expanded in K around 0 49.4%

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

       1. mul-1-neg 49.4%

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

       2. associate-+r- 49.4%

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

       3. distribute-lft-neg-in 49.4%

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

       4. associate--r+ 49.4%

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

       5. cancel-sign-sub 49.4%

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

   11. Simplified 49.4%

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

   12. Taylor expanded in l around inf 57.8%

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

       1. neg-mul-1 57.8%

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

   14. Simplified 57.8%

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

   if 1.85e-4 < l

   1. Initial program 73.5%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 l around inf 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in M around 0 98.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.000185:\\ \;\;\;\;e^{M \cdot \left(m + \left(n - M\right)\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in M around inf 45.8%

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

   1. associate-*r* 45.8%

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

   2. neg-mul-1 45.8%

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

   3. unpow2 45.8%

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

   4. sqr-neg 45.8%

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

   5. distribute-lft-out 47.7%

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

   6. unsub-neg 47.7%

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

5. Simplified 47.7%

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

6. Taylor expanded in m around inf 50.4%

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

   1. *-commutative 50.4%

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

   2. associate-*l* 50.4%

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

8. Simplified 50.4%

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

9. Taylor expanded in K around 0 55.9%

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

    1. mul-1-neg 55.9%

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

    2. associate-+r- 55.9%

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

    3. distribute-lft-neg-in 55.9%

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

    4. associate--r+ 55.9%

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

    5. cancel-sign-sub 55.9%

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

11. Simplified 55.9%

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

    1. sub-neg 55.9%

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

    2. add-sqr-sqrt 26.2%

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

    3. fabs-sqr 26.2%

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

    4. add-sqr-sqrt 73.1%

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

13. Applied egg-rr 73.1%

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

    1. sub-neg 73.1%

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

    2. associate--l- 73.1%

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

15. Simplified 73.1%

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

16. Final simplification 73.1%

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

Alternative 8: 35.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ e^{-\ell} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in l around inf 40.4%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
7. Step-by-step derivation

   1. mul-1-neg 40.4%

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

8. Simplified 40.4%

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

9. Taylor expanded in M around 0 38.8%

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

10. Final simplification 38.8%

    \[\leadsto e^{-\ell} \]
11. Add Preprocessing

Alternative 9: 6.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in l around inf 40.4%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
7. Step-by-step derivation

   1. mul-1-neg 40.4%

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

8. Simplified 40.4%

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

9. Taylor expanded in l around 0 8.8%

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

10. Final simplification 8.8%

    \[\leadsto \cos M \]
11. Add Preprocessing

Alternative 10: 6.7% accurate, 425.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in M around inf 51.1%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
7. Step-by-step derivation

   1. mul-1-neg 51.1%

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

8. Simplified 51.1%

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

9. Taylor expanded in M around 0 8.7%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 8.7%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024062 
(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)))))))