Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.5%

Time: 16.7s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((abs((n - m)) - (((((m + n) / 2.0d0) - m_1) ** 2.0d0) + l)))
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)) - (Math.pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. +-commutative 77.4%

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

   2. +-commutative 77.4%

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

   3. fabs-sub 77.4%

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

   4. associate-/l* 77.4%

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

   5. +-commutative 77.4%

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

3. Simplified 77.4%

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

4. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

6. Simplified 97.4%

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

7. Final simplification 97.4%

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

Alternative 2: 72.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -1.95 \cdot 10^{+21}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.85 \cdot 10^{-206}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\ell + \left(n \cdot n\right) \cdot 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -1.95e+21)
     (* (cos M) (exp (* -0.25 (* m m))))
     (if (<= m -1.85e-206)
       (* (cos M) (exp (- t_0 (+ l (* M M)))))
       (* (cos M) (exp (- t_0 (+ l (* (* n n) 0.25)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -1.95e+21) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (m <= -1.85e-206) {
		tmp = cos(M) * exp((t_0 - (l + (M * M))));
	} else {
		tmp = cos(M) * exp((t_0 - (l + ((n * n) * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-1.95d+21)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.85d-206)) then
        tmp = cos(m_1) * exp((t_0 - (l + (m_1 * m_1))))
    else
        tmp = cos(m_1) * exp((t_0 - (l + ((n * n) * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (m <= -1.95e+21) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.85e-206) {
		tmp = Math.cos(M) * Math.exp((t_0 - (l + (M * M))));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - (l + ((n * n) * 0.25))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -1.95e+21:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif m <= -1.85e-206:
		tmp = math.cos(M) * math.exp((t_0 - (l + (M * M))))
	else:
		tmp = math.cos(M) * math.exp((t_0 - (l + ((n * n) * 0.25))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -1.95e+21)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= -1.85e-206)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(l + Float64(M * M)))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(l + Float64(Float64(n * n) * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -1.95e+21)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (m <= -1.85e-206)
		tmp = cos(M) * exp((t_0 - (l + (M * M))));
	else
		tmp = cos(M) * exp((t_0 - (l + ((n * n) * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -1.95e+21], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.85e-206], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.95e21

   1. Initial program 74.1%

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

   2. Taylor expanded in m around inf 65.7%

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

      1. *-commutative 65.7%

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

      2. unpow2 65.7%

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

   4. Simplified 65.7%

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

   5. Taylor expanded in K around 0 86.4%

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

      1. cos-neg 86.4%

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

      2. fabs-sub 86.4%

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

      3. *-commutative 86.4%

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

      4. unpow2 86.4%

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

   7. Simplified 86.4%

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

   8. Taylor expanded in m around inf 98.3%

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

      1. unpow2 98.3%

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

   10. Simplified 98.3%

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

   if -1.95e21 < m < -1.84999999999999999e-206

   1. Initial program 82.5%

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

      1. +-commutative 82.5%

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

      2. +-commutative 82.5%

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

      3. fabs-sub 82.5%

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

      4. associate-/l* 85.0%

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

      5. +-commutative 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in M around inf 80.3%

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

      1. unpow2 80.3%

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

   9. Simplified 80.3%

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

   if -1.84999999999999999e-206 < m

   1. Initial program 77.3%

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

      1. +-commutative 77.3%

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

      2. +-commutative 77.3%

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

      3. fabs-sub 77.3%

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

      4. associate-/l* 76.7%

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

      5. +-commutative 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in n around inf 64.1%

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

      1. *-commutative 64.1%

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

      2. unpow2 64.1%

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

   9. Simplified 64.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.95 \cdot 10^{+21}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.85 \cdot 10^{-206}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + \left(n \cdot n\right) \cdot 0.25\right)}\\ \end{array} \]

Alternative 3: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{-82}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -5.6e-82)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= n 54.0)
     (* (cos M) (exp (- (fabs (- n m)) (+ l (* M M)))))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -5.6e-82) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp((fabs((n - m)) - (l + (M * M))));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-5.6d-82)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp((abs((n - m)) - (l + (m_1 * m_1))))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -5.6e-82) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((Math.abs((n - m)) - (l + (M * M))));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -5.6e-82:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp((math.fabs((n - m)) - (l + (M * M))))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -5.6e-82)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64(l + Float64(M * M)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -5.6e-82)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 54.0)
		tmp = cos(M) * exp((abs((n - m)) - (l + (M * M))));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -5.6e-82], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.60000000000000049e-82

   1. Initial program 74.7%

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

   2. Taylor expanded in m around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

   4. Simplified 37.5%

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

   5. Taylor expanded in K around 0 50.3%

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

      1. cos-neg 50.3%

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

      2. fabs-sub 50.3%

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

      3. *-commutative 50.3%

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

      4. unpow2 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in m around inf 53.2%

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

      1. unpow2 53.2%

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

   10. Simplified 53.2%

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

   if -5.60000000000000049e-82 < n < 54

   1. Initial program 83.1%

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

      1. +-commutative 83.1%

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

      2. +-commutative 83.1%

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

      3. fabs-sub 83.1%

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

      4. associate-/l* 83.1%

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

      5. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in M around inf 72.0%

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

      1. unpow2 72.0%

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

   9. Simplified 72.0%

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

   if 54 < n

   1. Initial program 71.9%

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

      1. +-commutative 71.9%

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

      2. +-commutative 71.9%

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

      3. fabs-sub 71.9%

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

      4. associate-/l* 71.9%

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

      5. +-commutative 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in n around inf 90.7%

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

      1. *-commutative 90.7%

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

      2. unpow2 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in n around inf 96.9%

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

       1. *-commutative 96.9%

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

       2. unpow2 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{-82}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 4: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 8.5 \cdot 10^{-197}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\ell + \left(m \cdot m\right) \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 8.5e-197)
     (* (cos M) (exp (- t_0 (+ l (* (* m m) 0.25)))))
     (if (<= n 54.0)
       (* (cos M) (exp (- t_0 (+ l (* M M)))))
       (* (cos M) (exp (* -0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 8.5e-197) {
		tmp = cos(M) * exp((t_0 - (l + ((m * m) * 0.25))));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp((t_0 - (l + (M * M))));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 8.5d-197) then
        tmp = cos(m_1) * exp((t_0 - (l + ((m * m) * 0.25d0))))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp((t_0 - (l + (m_1 * m_1))))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 8.5e-197) {
		tmp = Math.cos(M) * Math.exp((t_0 - (l + ((m * m) * 0.25))));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((t_0 - (l + (M * M))));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 8.5e-197:
		tmp = math.cos(M) * math.exp((t_0 - (l + ((m * m) * 0.25))))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp((t_0 - (l + (M * M))))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 8.5e-197)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(l + Float64(Float64(m * m) * 0.25)))));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(l + Float64(M * M)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 8.5e-197)
		tmp = cos(M) * exp((t_0 - (l + ((m * m) * 0.25))));
	elseif (n <= 54.0)
		tmp = cos(M) * exp((t_0 - (l + (M * M))));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 8.5e-197], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 8.5e-197

   1. Initial program 80.8%

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

   2. Taylor expanded in m around inf 52.7%

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

      1. *-commutative 52.7%

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

      2. unpow2 52.7%

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

   4. Simplified 52.7%

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

   5. Taylor expanded in K around 0 64.0%

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

      1. cos-neg 64.0%

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

      2. fabs-sub 64.0%

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

      3. *-commutative 64.0%

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

      4. unpow2 64.0%

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

   7. Simplified 64.0%

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

   if 8.5e-197 < n < 54

   1. Initial program 73.9%

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

      1. +-commutative 73.9%

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

      2. +-commutative 73.9%

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

      3. fabs-sub 73.9%

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

      4. associate-/l* 73.9%

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

      5. +-commutative 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in K around 0 91.0%

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

      1. cos-neg 91.0%

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

   6. Simplified 91.0%

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

   7. Taylor expanded in M around inf 72.3%

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

      1. unpow2 72.3%

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

   9. Simplified 72.3%

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

   if 54 < n

   1. Initial program 71.9%

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

      1. +-commutative 71.9%

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

      2. +-commutative 71.9%

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

      3. fabs-sub 71.9%

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

      4. associate-/l* 71.9%

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

      5. +-commutative 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in n around inf 90.7%

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

      1. *-commutative 90.7%

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

      2. unpow2 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in n around inf 96.9%

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

       1. *-commutative 96.9%

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

       2. unpow2 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.5 \cdot 10^{-197}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + \left(m \cdot m\right) \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.75 \cdot 10^{-9} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.75e-9) (not (<= M 27.0)))
   (* (cos M) (exp (* M (- M))))
   (* (cos M) (exp (* -0.25 (* m m))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.75e-9) || !(M <= 27.0)) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((-0.25 * (m * 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_1 <= (-1.75d-9)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (m * 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 <= -1.75e-9) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.75e-9) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.75e-9) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.75e-9) || ~((M <= 27.0)))
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((-0.25 * (m * m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.75e-9], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.75e-9 or 27 < M

   1. Initial program 79.0%

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

      1. +-commutative 79.0%

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

      2. +-commutative 79.0%

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

      3. fabs-sub 79.0%

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

      4. associate-/l* 79.0%

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

      5. +-commutative 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 84.1%

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

      1. unpow2 84.1%

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

   9. Simplified 84.1%

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

   10. Taylor expanded in M around inf 98.4%

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

       1. mul-1-neg 98.4%

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

       2. unpow2 98.4%

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

       3. distribute-rgt-neg-out 98.4%

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

   12. Simplified 98.4%

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

   if -1.75e-9 < M < 27

   1. Initial program 75.9%

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

   2. Taylor expanded in m around inf 54.3%

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

      1. *-commutative 54.3%

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

      2. unpow2 54.3%

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

   4. Simplified 54.3%

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

   5. Taylor expanded in K around 0 66.5%

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

      1. cos-neg 66.5%

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

      2. fabs-sub 66.5%

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

      3. *-commutative 66.5%

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

      4. unpow2 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in m around inf 59.9%

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

      1. unpow2 59.9%

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

   10. Simplified 59.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.75 \cdot 10^{-9} \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \]

Alternative 6: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.5 \cdot 10^{-172}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 3.5e-172)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= n 54.0)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3.5e-172) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 3.5d-172) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3.5e-172) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 3.5e-172:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 3.5e-172)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 3.5e-172)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 54.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 3.5e-172], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 3.50000000000000029e-172

   1. Initial program 81.5%

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

   2. Taylor expanded in m around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in K around 0 64.8%

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

      1. cos-neg 64.8%

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

      2. fabs-sub 64.8%

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

      3. *-commutative 64.8%

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

      4. unpow2 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in m around inf 59.8%

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

      1. unpow2 59.8%

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

   10. Simplified 59.8%

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

   if 3.50000000000000029e-172 < n < 54

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. +-commutative 69.6%

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

      3. fabs-sub 69.6%

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

      4. associate-/l* 72.4%

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

      5. +-commutative 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in K around 0 89.5%

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

      1. cos-neg 89.5%

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

   6. Simplified 89.5%

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

   7. Taylor expanded in M around inf 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   10. Taylor expanded in M around inf 60.0%

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

       1. mul-1-neg 60.0%

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

       2. unpow2 60.0%

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

       3. distribute-rgt-neg-out 60.0%

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

   12. Simplified 60.0%

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

   if 54 < n

   1. Initial program 71.9%

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

      1. +-commutative 71.9%

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

      2. +-commutative 71.9%

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

      3. fabs-sub 71.9%

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

      4. associate-/l* 71.9%

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

      5. +-commutative 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in n around inf 90.7%

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

      1. *-commutative 90.7%

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

      2. unpow2 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in n around inf 96.9%

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

       1. *-commutative 96.9%

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

       2. unpow2 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.5 \cdot 10^{-172}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 7: 70.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 2.5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -27.0) (not (<= M 2.5e-5)))
   (* (cos M) (exp (* M (- M))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 2.5e-5)) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-27.0d0)) .or. (.not. (m_1 <= 2.5d-5))) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 2.5e-5)) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -27.0) or not (M <= 2.5e-5):
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -27.0) || !(M <= 2.5e-5))
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -27.0) || ~((M <= 2.5e-5)))
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -27.0], N[Not[LessEqual[M, 2.5e-5]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 2.50000000000000012e-5 < M

   1. Initial program 80.6%

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

      1. +-commutative 80.6%

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

      2. +-commutative 80.6%

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

      3. fabs-sub 80.6%

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

      4. associate-/l* 80.6%

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

      5. +-commutative 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 84.1%

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

      1. unpow2 84.1%

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

   9. Simplified 84.1%

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

   10. Taylor expanded in M around inf 98.4%

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

       1. mul-1-neg 98.4%

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

       2. unpow2 98.4%

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

       3. distribute-rgt-neg-out 98.4%

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

   12. Simplified 98.4%

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

   if -27 < M < 2.50000000000000012e-5

   1. Initial program 74.4%

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

      1. +-commutative 74.4%

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

      2. +-commutative 74.4%

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

      3. fabs-sub 74.4%

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

      4. associate-/l* 74.4%

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

      5. +-commutative 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in K around 0 94.9%

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

      1. cos-neg 94.9%

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

   6. Simplified 94.9%

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

   7. Taylor expanded in M around inf 32.2%

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

      1. unpow2 32.2%

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

   9. Simplified 32.2%

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

   10. Taylor expanded in l around inf 41.8%

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

       1. mul-1-neg 41.8%

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

   12. Simplified 41.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 2.5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 8: 36.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. +-commutative 77.4%

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

   2. +-commutative 77.4%

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

   3. fabs-sub 77.4%

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

   4. associate-/l* 77.4%

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

   5. +-commutative 77.4%

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

3. Simplified 77.4%

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

4. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in M around inf 57.4%

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

   1. unpow2 57.4%

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

9. Simplified 57.4%

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

10. Taylor expanded in l around inf 33.5%

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

    1. mul-1-neg 33.5%

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

12. Simplified 33.5%

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

13. Final simplification 33.5%

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

Alternative 9: 35.7% 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 77.4%

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

   1. +-commutative 77.4%

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

   2. +-commutative 77.4%

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

   3. fabs-sub 77.4%

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

   4. associate-/l* 77.4%

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

   5. +-commutative 77.4%

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

3. Simplified 77.4%

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

4. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in M around inf 57.4%

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

   1. unpow2 57.4%

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

9. Simplified 57.4%

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

10. Taylor expanded in l around inf 33.5%

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

    1. mul-1-neg 33.5%

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

12. Simplified 33.5%

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

13. Taylor expanded in M around 0 33.5%

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

14. Final simplification 33.5%

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

Alternative 10: 7.4% 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 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. +-commutative 77.4%

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

   2. +-commutative 77.4%

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

   3. fabs-sub 77.4%

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

   4. associate-/l* 77.4%

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

   5. +-commutative 77.4%

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

3. Simplified 77.4%

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

4. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in M around inf 57.4%

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

   1. unpow2 57.4%

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

9. Simplified 57.4%

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

10. Taylor expanded in l around inf 33.5%

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

    1. mul-1-neg 33.5%

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

12. Simplified 33.5%

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

13. Taylor expanded in l around 0 6.7%

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

14. Final simplification 6.7%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023263 
(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)))))))