Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.7%

Time: 20.0s

Alternatives: 8

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% 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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- n m)) 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((n - m)) - 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((n - m)) - 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((n - m)) - 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((n - m)) - 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(n - m)) - 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((n - m)) - 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[(n - m), $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 77.1%

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

   1. associate-/l* 76.8%

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

   2. +-commutative 76.8%

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

   3. fabs-sub 76.8%

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

   4. +-commutative 76.8%

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

3. Simplified 76.8%

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

5. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

7. Simplified 97.4%

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

8. Final simplification 97.4%

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

Alternative 2: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -65000000000000:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -2.6 \cdot 10^{-280}:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-149} \lor \neg \left(m \leq 1.45 \cdot 10^{-87}\right):\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -65000000000000.0)
   (exp (* -0.25 (pow m 2.0)))
   (if (<= m -2.6e-280)
     (* (cos (- (/ (* m K) 2.0) M)) (exp (- l)))
     (if (or (<= m 3.3e-149) (not (<= m 1.45e-87)))
       (* (cos M) (exp (* -0.25 (pow n 2.0))))
       (* (cos M) (exp (- (pow M 2.0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -65000000000000.0) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else if (m <= -2.6e-280) {
		tmp = cos((((m * K) / 2.0) - M)) * exp(-l);
	} else if ((m <= 3.3e-149) || !(m <= 1.45e-87)) {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp(-pow(M, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-65000000000000.0d0)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= (-2.6d-280)) then
        tmp = cos((((m * k) / 2.0d0) - m_1)) * exp(-l)
    else if ((m <= 3.3d-149) .or. (.not. (m <= 1.45d-87))) then
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -65000000000000.0) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= -2.6e-280) {
		tmp = Math.cos((((m * K) / 2.0) - M)) * Math.exp(-l);
	} else if ((m <= 3.3e-149) || !(m <= 1.45e-87)) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -65000000000000.0:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= -2.6e-280:
		tmp = math.cos((((m * K) / 2.0) - M)) * math.exp(-l)
	elif (m <= 3.3e-149) or not (m <= 1.45e-87):
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -65000000000000.0)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	elseif (m <= -2.6e-280)
		tmp = Float64(cos(Float64(Float64(Float64(m * K) / 2.0) - M)) * exp(Float64(-l)));
	elseif ((m <= 3.3e-149) || !(m <= 1.45e-87))
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -65000000000000.0)
		tmp = exp((-0.25 * (m ^ 2.0)));
	elseif (m <= -2.6e-280)
		tmp = cos((((m * K) / 2.0) - M)) * exp(-l);
	elseif ((m <= 3.3e-149) || ~((m <= 1.45e-87)))
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -65000000000000.0], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -2.6e-280], N[(N[Cos[N[(N[(N[(m * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[m, 3.3e-149], N[Not[LessEqual[m, 1.45e-87]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -6.5e13

   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. Step-by-step derivation

      1. associate-/l* 73.0%

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

      2. +-commutative 73.0%

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

      3. fabs-sub 73.0%

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

      4. +-commutative 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 96.9%

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

   9. Taylor expanded in M around 0 96.9%

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

   if -6.5e13 < m < -2.6e-280

   1. Initial program 82.0%

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

   3. Taylor expanded in l around inf 44.5%

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

      1. mul-1-neg 44.5%

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

   5. Simplified 44.5%

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

   6. Taylor expanded in m around inf 53.4%

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

   if -2.6e-280 < m < 3.30000000000000017e-149 or 1.45e-87 < m

   1. Initial program 74.3%

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

   3. Taylor expanded in n around inf 42.9%

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

      1. *-commutative 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in K around 0 61.7%

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

      1. cos-neg 61.7%

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

   8. Simplified 61.7%

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

   if 3.30000000000000017e-149 < m < 1.45e-87

   1. Initial program 93.0%

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

      1. associate-/l* 93.8%

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

      2. +-commutative 93.8%

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

      3. fabs-sub 93.8%

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

      4. +-commutative 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in K around 0 93.0%

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

      1. cos-neg 93.0%

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

   7. Simplified 93.0%

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

   8. Taylor expanded in M around inf 51.5%

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

      1. mul-1-neg 51.5%

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

   10. Simplified 51.5%

       \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -65000000000000:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -2.6 \cdot 10^{-280}:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-149} \lor \neg \left(m \leq 1.45 \cdot 10^{-87}\right):\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+16}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m + \left(n \cdot 0.5 - M\right)\right) \cdot \left(M - n \cdot 0.5\right) - \left(\ell - \left|n - m\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1e+16)
   (exp (* -0.25 (pow m 2.0)))
   (*
    (cos M)
    (exp (- (* (+ m (- (* n 0.5) M)) (- M (* n 0.5))) (- l (fabs (- n m))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1e+16) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp((((m + ((n * 0.5) - M)) * (M - (n * 0.5))) - (l - fabs((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) :: tmp
    if (m <= (-1d+16)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp((((m + ((n * 0.5d0) - m_1)) * (m_1 - (n * 0.5d0))) - (l - abs((n - m)))))
    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+16) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp((((m + ((n * 0.5) - M)) * (M - (n * 0.5))) - (l - Math.abs((n - m)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1e+16:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp((((m + ((n * 0.5) - M)) * (M - (n * 0.5))) - (l - math.fabs((n - m)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1e+16)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(m + Float64(Float64(n * 0.5) - M)) * Float64(M - Float64(n * 0.5))) - Float64(l - abs(Float64(n - m))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1e+16)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp((((m + ((n * 0.5) - M)) * (M - (n * 0.5))) - (l - abs((n - m)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1e+16], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(m + N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l - N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1e16

   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. Step-by-step derivation

      1. associate-/l* 73.0%

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

      2. +-commutative 73.0%

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

      3. fabs-sub 73.0%

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

      4. +-commutative 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 96.9%

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

   9. Taylor expanded in M around 0 96.9%

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

   if -1e16 < 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. Step-by-step derivation

      1. associate-/l* 78.0%

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

      2. +-commutative 78.0%

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

      3. fabs-sub 78.0%

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

      4. +-commutative 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in K around 0 96.5%

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

      1. cos-neg 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in m around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. unpow2 84.2%

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

      3. distribute-rgt-out 88.4%

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

      4. *-commutative 88.4%

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

      5. *-commutative 88.4%

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

   10. Simplified 88.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+16}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m + \left(n \cdot 0.5 - M\right)\right) \cdot \left(M - n \cdot 0.5\right) - \left(\ell - \left|n - m\right|\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ \mathbf{if}\;m \leq -65000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -3.4 \cdot 10^{-280}:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0)))))
   (if (<= m -65000000000000.0)
     t_0
     (if (<= m -3.4e-280)
       (* (cos (- (/ (* m K) 2.0) M)) (exp (- l)))
       (if (<= m 53.0) (* (cos M) (exp (- (pow M 2.0)))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double tmp;
	if (m <= -65000000000000.0) {
		tmp = t_0;
	} else if (m <= -3.4e-280) {
		tmp = cos((((m * K) / 2.0) - M)) * exp(-l);
	} else if (m <= 53.0) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (m ** 2.0d0)))
    if (m <= (-65000000000000.0d0)) then
        tmp = t_0
    else if (m <= (-3.4d-280)) then
        tmp = cos((((m * k) / 2.0d0) - m_1)) * exp(-l)
    else if (m <= 53.0d0) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(m, 2.0)));
	double tmp;
	if (m <= -65000000000000.0) {
		tmp = t_0;
	} else if (m <= -3.4e-280) {
		tmp = Math.cos((((m * K) / 2.0) - M)) * Math.exp(-l);
	} else if (m <= 53.0) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	tmp = 0
	if m <= -65000000000000.0:
		tmp = t_0
	elif m <= -3.4e-280:
		tmp = math.cos((((m * K) / 2.0) - M)) * math.exp(-l)
	elif m <= 53.0:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	tmp = 0.0
	if (m <= -65000000000000.0)
		tmp = t_0;
	elseif (m <= -3.4e-280)
		tmp = Float64(cos(Float64(Float64(Float64(m * K) / 2.0) - M)) * exp(Float64(-l)));
	elseif (m <= 53.0)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	tmp = 0.0;
	if (m <= -65000000000000.0)
		tmp = t_0;
	elseif (m <= -3.4e-280)
		tmp = cos((((m * K) / 2.0) - M)) * exp(-l);
	elseif (m <= 53.0)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -65000000000000.0], t$95$0, If[LessEqual[m, -3.4e-280], N[(N[Cos[N[(N[(N[(m * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 53.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.5e13 or 53 < m

   1. Initial program 69.7%

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

      1. associate-/l* 69.7%

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

      2. +-commutative 69.7%

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

      3. fabs-sub 69.7%

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

      4. +-commutative 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 97.6%

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

   9. Taylor expanded in M around 0 97.6%

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

   if -6.5e13 < m < -3.3999999999999998e-280

   1. Initial program 82.0%

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

   3. Taylor expanded in l around inf 44.5%

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

      1. mul-1-neg 44.5%

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

   5. Simplified 44.5%

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

   6. Taylor expanded in m around inf 53.4%

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

   if -3.3999999999999998e-280 < m < 53

   1. Initial program 86.0%

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

      1. associate-/l* 86.1%

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

      2. +-commutative 86.1%

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

      3. fabs-sub 86.1%

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

      4. +-commutative 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in M around inf 63.5%

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

      1. mul-1-neg 63.5%

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

   10. Simplified 63.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -65000000000000:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -3.4 \cdot 10^{-280}:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-\ell}\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -2.4e-37)
   (* (cos M) (exp l))
   (if (<= l 5.4e-11) (exp (* -0.25 (pow m 2.0))) (/ (cos M) (exp l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.4e-37) {
		tmp = cos(M) * exp(l);
	} else if (l <= 5.4e-11) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2.4d-37)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 5.4d-11) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.4e-37) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 5.4e-11) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -2.4e-37:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 5.4e-11:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -2.4e-37)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 5.4e-11)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -2.4e-37)
		tmp = cos(M) * exp(l);
	elseif (l <= 5.4e-11)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -2.4e-37], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.4e-11], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.39999999999999991e-37

   1. Initial program 71.5%

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

      1. associate-/l* 70.0%

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

      2. +-commutative 70.0%

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

      3. fabs-sub 70.0%

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

      4. +-commutative 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in l around inf 25.5%

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

      1. neg-mul-1 25.5%

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

   10. Simplified 25.5%

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

       1. expm1-log1p-u 23.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 23.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 23.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 23.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 23.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 69.1%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 69.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 69.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 69.1%

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

   14. Simplified 69.1%

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

   if -2.39999999999999991e-37 < l < 5.40000000000000009e-11

   1. Initial program 80.3%

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

      1. associate-/l* 80.4%

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

      2. +-commutative 80.4%

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

      3. fabs-sub 80.4%

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

      4. +-commutative 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in m around inf 64.0%

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

   9. Taylor expanded in M around 0 64.0%

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

   if 5.40000000000000009e-11 < l

   1. Initial program 77.4%

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

      1. associate-/l* 77.4%

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

      2. +-commutative 77.4%

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

      3. fabs-sub 77.4%

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

      4. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in l around inf 95.8%

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

      1. neg-mul-1 95.8%

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

   10. Simplified 95.8%

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

       1. exp-neg 95.8%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

       2. un-div-inv 95.8%

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

   12. Applied egg-rr 95.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-5}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1e-5) (* (cos M) (exp l)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1e-5) {
		tmp = cos(M) * exp(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 <= 1d-5) then
        tmp = cos(m_1) * exp(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 <= 1e-5) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1e-5:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1e-5)
		tmp = Float64(cos(M) * exp(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 <= 1e-5)
		tmp = cos(M) * exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1e-5], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000008e-5

   1. Initial program 76.6%

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

      1. associate-/l* 76.1%

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

      2. +-commutative 76.1%

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

      3. fabs-sub 76.1%

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

      4. +-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in l around inf 17.7%

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

      1. neg-mul-1 17.7%

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

   10. Simplified 17.7%

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

       1. expm1-log1p-u 17.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 17.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 12.6%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 17.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 17.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 4.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 34.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 34.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 34.0%

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

   14. Simplified 34.0%

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

   if 1.00000000000000008e-5 < l

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. +-commutative 78.6%

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

      3. fabs-sub 78.6%

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

      4. +-commutative 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in l around inf 97.2%

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

      1. neg-mul-1 97.2%

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

   10. Simplified 97.2%

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

   11. Taylor expanded in M around 0 97.2%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-5}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -5e-64) (* (cos M) (exp l)) (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -5e-64) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-64)) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -5e-64) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -5e-64:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -5e-64)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -5e-64)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -5e-64], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.00000000000000033e-64

   1. Initial program 73.4%

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

      1. associate-/l* 72.2%

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

      2. +-commutative 72.2%

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

      3. fabs-sub 72.2%

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

      4. +-commutative 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in l around inf 22.9%

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

      1. neg-mul-1 22.9%

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

   10. Simplified 22.9%

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

       1. expm1-log1p-u 21.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 21.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 21.3%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 21.3%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 21.3%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 61.6%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 61.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 61.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 61.6%

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

   14. Simplified 61.6%

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

   if -5.00000000000000033e-64 < l

   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. Step-by-step derivation

      1. associate-/l* 78.8%

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

      2. +-commutative 78.8%

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

      3. fabs-sub 78.8%

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

      4. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in l around inf 46.3%

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

      1. neg-mul-1 46.3%

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

   10. Simplified 46.3%

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

       1. exp-neg 46.3%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

       2. un-div-inv 46.3%

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

   12. Applied egg-rr 46.3%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.9% 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 77.1%

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

   1. associate-/l* 76.8%

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

   2. +-commutative 76.8%

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

   3. fabs-sub 76.8%

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

   4. +-commutative 76.8%

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

3. Simplified 76.8%

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

5. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in l around inf 39.1%

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

   1. neg-mul-1 39.1%

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

10. Simplified 39.1%

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

11. Taylor expanded in M around 0 39.1%

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

12. Final simplification 39.1%

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

Reproduce

herbie shell --seed 2024011 
(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)))))))