Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.5%

Time: 16.6s

Alternatives: 13

Speedup: 1.0×

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: 76.3% 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

\[\begin{array}{l} \\ \cos M \cdot e^{\left|m - n\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- m n)) (+ l (pow (- (* (+ m n) 0.5) M) 2.0))))))

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((abs((m - n)) - (l + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(m - n)) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. associate-/l* 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. associate--r- 69.7%

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

3. Simplified 69.7%

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

4. Taylor expanded in K around 0 96.3%

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

6. Simplified 96.3%

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

7. Final simplification 96.3%

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

Alternative 2: 73.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 68000:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|m - n\right|\right) + \left(m \cdot m\right) \cdot 0.25}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 68000.0)
   (/ (cos M) (exp (+ (- l (fabs (- m n))) (* (* m m) 0.25))))
   (pow (exp n) (* n -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 68000.0) {
		tmp = cos(M) / exp(((l - fabs((m - n))) + ((m * m) * 0.25)));
	} else {
		tmp = pow(exp(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) :: tmp
    if (n <= 68000.0d0) then
        tmp = cos(m_1) / exp(((l - abs((m - n))) + ((m * m) * 0.25d0)))
    else
        tmp = exp(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 tmp;
	if (n <= 68000.0) {
		tmp = Math.cos(M) / Math.exp(((l - Math.abs((m - n))) + ((m * m) * 0.25)));
	} else {
		tmp = Math.pow(Math.exp(n), (n * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 68000.0:
		tmp = math.cos(M) / math.exp(((l - math.fabs((m - n))) + ((m * m) * 0.25)))
	else:
		tmp = math.pow(math.exp(n), (n * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 68000.0)
		tmp = Float64(cos(M) / exp(Float64(Float64(l - abs(Float64(m - n))) + Float64(Float64(m * m) * 0.25))));
	else
		tmp = exp(n) ^ Float64(n * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 68000.0)
		tmp = cos(M) / exp(((l - abs((m - n))) + ((m * m) * 0.25)));
	else
		tmp = exp(n) ^ (n * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 68000.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 68000

   1. Initial program 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in m around inf 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in K around 0 68.7%

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

      1. cos-neg 38.1%

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

   9. Simplified 68.7%

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

   if 68000 < n

   1. Initial program 57.6%

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

      1. associate-/l* 57.6%

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

      2. associate--r- 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in K around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r* 77.6%

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

   9. Simplified 77.6%

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

   10. Taylor expanded in n around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow2 97.0%

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

       3. associate-*l* 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in M around 0 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow2 97.0%

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

       3. associate-*r* 97.0%

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

       4. exp-prod 97.0%

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

   15. Simplified 97.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 68000:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|m - n\right|\right) + \left(m \cdot m\right) \cdot 0.25}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 3: 68.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{if}\;n \leq -55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-295}:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)}{e^{\ell}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{n \cdot \left(n \cdot 0.25\right) + n \cdot \left(\left(m \cdot 0.5 - M\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (pow (exp n) (* n -0.25))))
   (if (<= n -55.0)
     t_0
     (if (<= n -7.6e-295)
       (exp (* m (- (+ (* n 0.5) 1.0) M)))
       (if (<= n 4.3e-187)
         (/ (cos (- (* 0.5 (* n K)) M)) (exp l))
         (if (<= n 54.0)
           (exp (+ (* n (* n 0.25)) (* n (+ (- (* m 0.5) M) -1.0))))
           t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(exp(n), (n * -0.25));
	double tmp;
	if (n <= -55.0) {
		tmp = t_0;
	} else if (n <= -7.6e-295) {
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	} else if (n <= 4.3e-187) {
		tmp = cos(((0.5 * (n * K)) - M)) / exp(l);
	} else if (n <= 54.0) {
		tmp = exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))));
	} 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(n) ** (n * (-0.25d0))
    if (n <= (-55.0d0)) then
        tmp = t_0
    else if (n <= (-7.6d-295)) then
        tmp = exp((m * (((n * 0.5d0) + 1.0d0) - m_1)))
    else if (n <= 4.3d-187) then
        tmp = cos(((0.5d0 * (n * k)) - m_1)) / exp(l)
    else if (n <= 54.0d0) then
        tmp = exp(((n * (n * 0.25d0)) + (n * (((m * 0.5d0) - m_1) + (-1.0d0)))))
    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.pow(Math.exp(n), (n * -0.25));
	double tmp;
	if (n <= -55.0) {
		tmp = t_0;
	} else if (n <= -7.6e-295) {
		tmp = Math.exp((m * (((n * 0.5) + 1.0) - M)));
	} else if (n <= 4.3e-187) {
		tmp = Math.cos(((0.5 * (n * K)) - M)) / Math.exp(l);
	} else if (n <= 54.0) {
		tmp = Math.exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(math.exp(n), (n * -0.25))
	tmp = 0
	if n <= -55.0:
		tmp = t_0
	elif n <= -7.6e-295:
		tmp = math.exp((m * (((n * 0.5) + 1.0) - M)))
	elif n <= 4.3e-187:
		tmp = math.cos(((0.5 * (n * K)) - M)) / math.exp(l)
	elif n <= 54.0:
		tmp = math.exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(n) ^ Float64(n * -0.25)
	tmp = 0.0
	if (n <= -55.0)
		tmp = t_0;
	elseif (n <= -7.6e-295)
		tmp = exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M)));
	elseif (n <= 4.3e-187)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(n * K)) - M)) / exp(l));
	elseif (n <= 54.0)
		tmp = exp(Float64(Float64(n * Float64(n * 0.25)) + Float64(n * Float64(Float64(Float64(m * 0.5) - M) + -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(n) ^ (n * -0.25);
	tmp = 0.0;
	if (n <= -55.0)
		tmp = t_0;
	elseif (n <= -7.6e-295)
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	elseif (n <= 4.3e-187)
		tmp = cos(((0.5 * (n * K)) - M)) / exp(l);
	elseif (n <= 54.0)
		tmp = exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -55.0], t$95$0, If[LessEqual[n, -7.6e-295], N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.3e-187], N[(N[Cos[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[N[(N[(n * N[(n * 0.25), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -55 or 54 < n

   1. Initial program 63.3%

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

      1. associate-/l* 63.3%

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

      2. associate--r- 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in K around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 83.1%

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

      1. unpow2 83.1%

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

      2. associate-*r* 83.1%

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

   9. Simplified 83.1%

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

   10. Taylor expanded in n around inf 97.7%

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

       1. *-commutative 97.7%

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

       2. unpow2 97.7%

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

       3. associate-*l* 97.7%

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

   12. Simplified 97.7%

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

   13. Taylor expanded in M around 0 97.7%

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

       1. *-commutative 97.7%

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

       2. unpow2 97.7%

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

       3. associate-*r* 97.7%

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

       4. exp-prod 97.7%

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

   15. Simplified 97.7%

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

   if -55 < n < -7.60000000000000037e-295

   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. associate--r- 78.8%

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

   3. Simplified 78.8%

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

      1. add-exp-log 75.7%

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

      2. *-commutative 75.7%

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

      3. log-prod 49.5%

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

   5. Applied egg-rr 23.7%

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

   6. Taylor expanded in K around 0 26.2%

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

      1. associate--l+ 26.2%

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

      2. +-commutative 26.2%

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

      3. +-commutative 26.2%

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

      4. cos-neg 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in m around inf 23.2%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 23.2%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 23.2%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in m around 0 49.1%

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

   if -7.60000000000000037e-295 < n < 4.3e-187

   1. Initial program 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in l around inf 56.8%

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

   5. Taylor expanded in m around 0 61.9%

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

   if 4.3e-187 < n < 54

   1. Initial program 72.0%

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

      1. associate-/l* 71.6%

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

      2. associate--r- 71.6%

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

   3. Simplified 71.6%

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

      1. add-exp-log 71.6%

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

      2. *-commutative 71.6%

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

      3. log-prod 47.9%

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

   5. Applied egg-rr 13.5%

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

   6. Taylor expanded in K around 0 14.5%

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

      1. associate--l+ 14.5%

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

      2. +-commutative 14.5%

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

      3. +-commutative 14.5%

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

      4. cos-neg 14.5%

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

   8. Simplified 14.5%

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

   9. Taylor expanded in n around -inf 32.2%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)\right) + 0.25 \cdot {n}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative 32.2%

          \[\leadsto e^{\color{blue}{0.25 \cdot {n}^{2} + -1 \cdot \left(n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)\right)}} \]

       2. mul-1-neg 32.2%

          \[\leadsto e^{0.25 \cdot {n}^{2} + \color{blue}{\left(-n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)\right)}} \]

       3. unsub-neg 32.2%

          \[\leadsto e^{\color{blue}{0.25 \cdot {n}^{2} - n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)}} \]

       4. unpow2 32.2%

          \[\leadsto e^{0.25 \cdot \color{blue}{\left(n \cdot n\right)} - n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)} \]

       5. associate-*r* 32.2%

          \[\leadsto e^{\color{blue}{\left(0.25 \cdot n\right) \cdot n} - n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)} \]

       6. mul-1-neg 32.2%

          \[\leadsto e^{\left(0.25 \cdot n\right) \cdot n - n \cdot \left(1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}\right)} \]

       7. unsub-neg 32.2%

          \[\leadsto e^{\left(0.25 \cdot n\right) \cdot n - n \cdot \color{blue}{\left(1 - \left(0.5 \cdot m - M\right)\right)}} \]

       8. *-commutative 32.2%

          \[\leadsto e^{\left(0.25 \cdot n\right) \cdot n - n \cdot \left(1 - \left(\color{blue}{m \cdot 0.5} - M\right)\right)} \]

   11. Simplified 32.2%

       \[\leadsto e^{\color{blue}{\left(0.25 \cdot n\right) \cdot n - n \cdot \left(1 - \left(m \cdot 0.5 - M\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -55:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-295}:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)}{e^{\ell}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{n \cdot \left(n \cdot 0.25\right) + n \cdot \left(\left(m \cdot 0.5 - M\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 4: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left|m - n\right| - \left(m \cdot m\right) \cdot 0.25}\\ \mathbf{if}\;n \leq 5.6 \cdot 10^{-289}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (fabs (- m n)) (* (* m m) 0.25)))))
   (if (<= n 5.6e-289)
     t_0
     (if (<= n 7.5e-224)
       (/ (cos M) (exp l))
       (if (<= n 54.0) t_0 (pow (exp n) (* n -0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((fabs((m - n)) - ((m * m) * 0.25)));
	double tmp;
	if (n <= 5.6e-289) {
		tmp = t_0;
	} else if (n <= 7.5e-224) {
		tmp = cos(M) / exp(l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = pow(exp(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 = exp((abs((m - n)) - ((m * m) * 0.25d0)))
    if (n <= 5.6d-289) then
        tmp = t_0
    else if (n <= 7.5d-224) then
        tmp = cos(m_1) / exp(l)
    else if (n <= 54.0d0) then
        tmp = t_0
    else
        tmp = exp(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.exp((Math.abs((m - n)) - ((m * m) * 0.25)));
	double tmp;
	if (n <= 5.6e-289) {
		tmp = t_0;
	} else if (n <= 7.5e-224) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (n <= 54.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow(Math.exp(n), (n * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((math.fabs((m - n)) - ((m * m) * 0.25)))
	tmp = 0
	if n <= 5.6e-289:
		tmp = t_0
	elif n <= 7.5e-224:
		tmp = math.cos(M) / math.exp(l)
	elif n <= 54.0:
		tmp = t_0
	else:
		tmp = math.pow(math.exp(n), (n * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(abs(Float64(m - n)) - Float64(Float64(m * m) * 0.25)))
	tmp = 0.0
	if (n <= 5.6e-289)
		tmp = t_0;
	elseif (n <= 7.5e-224)
		tmp = Float64(cos(M) / exp(l));
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(n) ^ Float64(n * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((abs((m - n)) - ((m * m) * 0.25)));
	tmp = 0.0;
	if (n <= 5.6e-289)
		tmp = t_0;
	elseif (n <= 7.5e-224)
		tmp = cos(M) / exp(l);
	elseif (n <= 54.0)
		tmp = t_0;
	else
		tmp = exp(n) ^ (n * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 5.6e-289], t$95$0, If[LessEqual[n, 7.5e-224], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], t$95$0, N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.5999999999999997e-289 or 7.49999999999999978e-224 < n < 54

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in m around inf 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in K around 0 67.3%

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

      1. cos-neg 36.3%

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

   9. Simplified 67.3%

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

   10. Taylor expanded in M around 0 66.7%

       \[\leadsto \color{blue}{\frac{1}{e^{\left(\ell + 0.25 \cdot {m}^{2}\right) - \left|n - m\right|}}} \]
   11. Step-by-step derivation

       1. rec-exp 66.7%

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

       2. associate--l+ 66.7%

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

       3. associate--l+ 66.7%

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

       4. unpow2 66.7%

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

   12. Simplified 66.7%

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

   13. Taylor expanded in l around 0 55.7%

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

       1. unpow2 55.7%

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

       2. fabs-sub 55.7%

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

   15. Simplified 55.7%

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

   if 5.5999999999999997e-289 < n < 7.49999999999999978e-224

   1. Initial program 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in l around inf 64.3%

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

   5. Taylor expanded in K around 0 66.3%

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

      1. cos-neg 66.3%

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

   7. Simplified 66.3%

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

   if 54 < n

   1. Initial program 57.6%

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

      1. associate-/l* 57.6%

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

      2. associate--r- 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in K around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r* 77.6%

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

   9. Simplified 77.6%

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

   10. Taylor expanded in n around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow2 97.0%

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

       3. associate-*l* 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in M around 0 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow2 97.0%

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

       3. associate-*r* 97.0%

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

       4. exp-prod 97.0%

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

   15. Simplified 97.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-289}:\\ \;\;\;\;e^{\left|m - n\right| - \left(m \cdot m\right) \cdot 0.25}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{\left|m - n\right| - \left(m \cdot m\right) \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 5: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 68000:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + \left(m \cdot m\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 68000.0)
   (exp (- (fabs (- m n)) (+ l (* (* m m) 0.25))))
   (pow (exp n) (* n -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 68000.0) {
		tmp = exp((fabs((m - n)) - (l + ((m * m) * 0.25))));
	} else {
		tmp = pow(exp(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) :: tmp
    if (n <= 68000.0d0) then
        tmp = exp((abs((m - n)) - (l + ((m * m) * 0.25d0))))
    else
        tmp = exp(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 tmp;
	if (n <= 68000.0) {
		tmp = Math.exp((Math.abs((m - n)) - (l + ((m * m) * 0.25))));
	} else {
		tmp = Math.pow(Math.exp(n), (n * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 68000.0:
		tmp = math.exp((math.fabs((m - n)) - (l + ((m * m) * 0.25))))
	else:
		tmp = math.pow(math.exp(n), (n * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 68000.0)
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(Float64(m * m) * 0.25))));
	else
		tmp = exp(n) ^ Float64(n * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 68000.0)
		tmp = exp((abs((m - n)) - (l + ((m * m) * 0.25))));
	else
		tmp = exp(n) ^ (n * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 68000.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 68000

   1. Initial program 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in m around inf 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in K around 0 68.7%

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

      1. cos-neg 38.1%

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

   9. Simplified 68.7%

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

   10. Taylor expanded in M around 0 68.2%

       \[\leadsto \color{blue}{\frac{1}{e^{\left(\ell + 0.25 \cdot {m}^{2}\right) - \left|n - m\right|}}} \]
   11. Step-by-step derivation

       1. rec-exp 68.2%

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

       2. associate--l+ 68.2%

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

       3. associate--l+ 68.2%

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

       4. unpow2 68.2%

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

   12. Simplified 68.2%

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

   if 68000 < n

   1. Initial program 57.6%

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

      1. associate-/l* 57.6%

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

      2. associate--r- 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in K around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r* 77.6%

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

   9. Simplified 77.6%

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

   10. Taylor expanded in n around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow2 97.0%

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

       3. associate-*l* 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in M around 0 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow2 97.0%

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

       3. associate-*r* 97.0%

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

       4. exp-prod 97.0%

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

   15. Simplified 97.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 68000:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + \left(m \cdot m\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 6: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{if}\;n \leq -52:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-295}:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{n \cdot \left(n \cdot 0.25\right) + n \cdot \left(\left(m \cdot 0.5 - M\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (pow (exp n) (* n -0.25))))
   (if (<= n -52.0)
     t_0
     (if (<= n -7.6e-295)
       (exp (* m (- (+ (* n 0.5) 1.0) M)))
       (if (<= n 1.4e-180)
         (/ (cos M) (exp l))
         (if (<= n 54.0)
           (exp (+ (* n (* n 0.25)) (* n (+ (- (* m 0.5) M) -1.0))))
           t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(exp(n), (n * -0.25));
	double tmp;
	if (n <= -52.0) {
		tmp = t_0;
	} else if (n <= -7.6e-295) {
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	} else if (n <= 1.4e-180) {
		tmp = cos(M) / exp(l);
	} else if (n <= 54.0) {
		tmp = exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))));
	} 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(n) ** (n * (-0.25d0))
    if (n <= (-52.0d0)) then
        tmp = t_0
    else if (n <= (-7.6d-295)) then
        tmp = exp((m * (((n * 0.5d0) + 1.0d0) - m_1)))
    else if (n <= 1.4d-180) then
        tmp = cos(m_1) / exp(l)
    else if (n <= 54.0d0) then
        tmp = exp(((n * (n * 0.25d0)) + (n * (((m * 0.5d0) - m_1) + (-1.0d0)))))
    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.pow(Math.exp(n), (n * -0.25));
	double tmp;
	if (n <= -52.0) {
		tmp = t_0;
	} else if (n <= -7.6e-295) {
		tmp = Math.exp((m * (((n * 0.5) + 1.0) - M)));
	} else if (n <= 1.4e-180) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (n <= 54.0) {
		tmp = Math.exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(math.exp(n), (n * -0.25))
	tmp = 0
	if n <= -52.0:
		tmp = t_0
	elif n <= -7.6e-295:
		tmp = math.exp((m * (((n * 0.5) + 1.0) - M)))
	elif n <= 1.4e-180:
		tmp = math.cos(M) / math.exp(l)
	elif n <= 54.0:
		tmp = math.exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(n) ^ Float64(n * -0.25)
	tmp = 0.0
	if (n <= -52.0)
		tmp = t_0;
	elseif (n <= -7.6e-295)
		tmp = exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M)));
	elseif (n <= 1.4e-180)
		tmp = Float64(cos(M) / exp(l));
	elseif (n <= 54.0)
		tmp = exp(Float64(Float64(n * Float64(n * 0.25)) + Float64(n * Float64(Float64(Float64(m * 0.5) - M) + -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(n) ^ (n * -0.25);
	tmp = 0.0;
	if (n <= -52.0)
		tmp = t_0;
	elseif (n <= -7.6e-295)
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	elseif (n <= 1.4e-180)
		tmp = cos(M) / exp(l);
	elseif (n <= 54.0)
		tmp = exp(((n * (n * 0.25)) + (n * (((m * 0.5) - M) + -1.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -52.0], t$95$0, If[LessEqual[n, -7.6e-295], N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.4e-180], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[N[(N[(n * N[(n * 0.25), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -52 or 54 < n

   1. Initial program 63.3%

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

      1. associate-/l* 63.3%

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

      2. associate--r- 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in K around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 83.1%

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

      1. unpow2 83.1%

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

      2. associate-*r* 83.1%

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

   9. Simplified 83.1%

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

   10. Taylor expanded in n around inf 97.7%

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

       1. *-commutative 97.7%

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

       2. unpow2 97.7%

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

       3. associate-*l* 97.7%

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

   12. Simplified 97.7%

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

   13. Taylor expanded in M around 0 97.7%

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

       1. *-commutative 97.7%

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

       2. unpow2 97.7%

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

       3. associate-*r* 97.7%

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

       4. exp-prod 97.7%

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

   15. Simplified 97.7%

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

   if -52 < n < -7.60000000000000037e-295

   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. associate--r- 78.8%

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

   3. Simplified 78.8%

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

      1. add-exp-log 75.7%

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

      2. *-commutative 75.7%

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

      3. log-prod 49.5%

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

   5. Applied egg-rr 23.7%

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

   6. Taylor expanded in K around 0 26.2%

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

      1. associate--l+ 26.2%

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

      2. +-commutative 26.2%

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

      3. +-commutative 26.2%

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

      4. cos-neg 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in m around inf 23.2%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 23.2%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 23.2%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 23.2%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in m around 0 49.1%

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

   if -7.60000000000000037e-295 < n < 1.39999999999999999e-180

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in l around inf 50.4%

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

   5. Taylor expanded in K around 0 52.9%

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

      1. cos-neg 52.9%

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

   7. Simplified 52.9%

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

   if 1.39999999999999999e-180 < n < 54

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. associate--r- 73.6%

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

   3. Simplified 73.6%

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

      1. add-exp-log 73.6%

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

      2. *-commutative 73.6%

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

      3. log-prod 50.0%

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

   5. Applied egg-rr 14.0%

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

   6. Taylor expanded in K around 0 14.4%

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

      1. associate--l+ 14.4%

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

      2. +-commutative 14.4%

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

      3. +-commutative 14.4%

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

      4. cos-neg 14.4%

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

   8. Simplified 14.4%

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

   9. Taylor expanded in n around -inf 34.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)\right) + 0.25 \cdot {n}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative 34.1%

          \[\leadsto e^{\color{blue}{0.25 \cdot {n}^{2} + -1 \cdot \left(n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)\right)}} \]

       2. mul-1-neg 34.1%

          \[\leadsto e^{0.25 \cdot {n}^{2} + \color{blue}{\left(-n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)\right)}} \]

       3. unsub-neg 34.1%

          \[\leadsto e^{\color{blue}{0.25 \cdot {n}^{2} - n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)}} \]

       4. unpow2 34.1%

          \[\leadsto e^{0.25 \cdot \color{blue}{\left(n \cdot n\right)} - n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)} \]

       5. associate-*r* 34.1%

          \[\leadsto e^{\color{blue}{\left(0.25 \cdot n\right) \cdot n} - n \cdot \left(1 + -1 \cdot \left(0.5 \cdot m - M\right)\right)} \]

       6. mul-1-neg 34.1%

          \[\leadsto e^{\left(0.25 \cdot n\right) \cdot n - n \cdot \left(1 + \color{blue}{\left(-\left(0.5 \cdot m - M\right)\right)}\right)} \]

       7. unsub-neg 34.1%

          \[\leadsto e^{\left(0.25 \cdot n\right) \cdot n - n \cdot \color{blue}{\left(1 - \left(0.5 \cdot m - M\right)\right)}} \]

       8. *-commutative 34.1%

          \[\leadsto e^{\left(0.25 \cdot n\right) \cdot n - n \cdot \left(1 - \left(\color{blue}{m \cdot 0.5} - M\right)\right)} \]

   11. Simplified 34.1%

       \[\leadsto e^{\color{blue}{\left(0.25 \cdot n\right) \cdot n - n \cdot \left(1 - \left(m \cdot 0.5 - M\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -52:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-295}:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{n \cdot \left(n \cdot 0.25\right) + n \cdot \left(\left(m \cdot 0.5 - M\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 7: 55.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.8:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.8) (exp (* m (- (+ (* n 0.5) 1.0) M))) (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.8) {
		tmp = exp((m * (((n * 0.5) + 1.0) - 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 (l <= 0.8d0) then
        tmp = exp((m * (((n * 0.5d0) + 1.0d0) - 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 (l <= 0.8) {
		tmp = Math.exp((m * (((n * 0.5) + 1.0) - M)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.8:
		tmp = math.exp((m * (((n * 0.5) + 1.0) - M)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.8)
		tmp = exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M)));
	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 <= 0.8)
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.8], N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.80000000000000004

   1. Initial program 69.1%

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

      1. associate-/l* 69.0%

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

      2. associate--r- 69.0%

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

   3. Simplified 69.0%

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

      1. add-exp-log 67.9%

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

      2. *-commutative 67.9%

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

      3. log-prod 38.3%

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

   5. Applied egg-rr 11.3%

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

   6. Taylor expanded in K around 0 12.9%

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

      1. associate--l+ 12.9%

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

      2. +-commutative 12.9%

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

      3. +-commutative 12.9%

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

      4. cos-neg 12.9%

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

   8. Simplified 12.9%

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

   9. Taylor expanded in m around inf 24.6%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 24.6%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 24.6%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in m around 0 44.6%

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

   if 0.80000000000000004 < l

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in l around inf 71.0%

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

   5. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.8:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 8: 55.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.75:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.75) (exp (* m (- (+ (* n 0.5) 1.0) M))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.75) {
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 0.75d0) then
        tmp = exp((m * (((n * 0.5d0) + 1.0d0) - m_1)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.75) {
		tmp = Math.exp((m * (((n * 0.5) + 1.0) - M)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.75:
		tmp = math.exp((m * (((n * 0.5) + 1.0) - M)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.75)
		tmp = exp(Float64(m * Float64(Float64(Float64(n * 0.5) + 1.0) - M)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.75)
		tmp = exp((m * (((n * 0.5) + 1.0) - M)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.75], N[Exp[N[(m * N[(N[(N[(n * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.75

   1. Initial program 69.1%

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

      1. associate-/l* 69.0%

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

      2. associate--r- 69.0%

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

   3. Simplified 69.0%

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

      1. add-exp-log 67.9%

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

      2. *-commutative 67.9%

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

      3. log-prod 38.3%

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

   5. Applied egg-rr 11.3%

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

   6. Taylor expanded in K around 0 12.9%

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

      1. associate--l+ 12.9%

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

      2. +-commutative 12.9%

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

      3. +-commutative 12.9%

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

      4. cos-neg 12.9%

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

   8. Simplified 12.9%

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

   9. Taylor expanded in m around inf 24.6%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 24.6%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 24.6%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in m around 0 44.6%

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

   if 0.75 < l

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in l around inf 71.0%

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

   5. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in M around 0 97.9%

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

      1. rec-exp 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.75:\\ \;\;\;\;e^{m \cdot \left(\left(n \cdot 0.5 + 1\right) - M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 9: 48.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-300}:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 0.004:\\ \;\;\;\;e^{M \cdot \left(-m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 4.2e-300)
   (exp (* n (* m 0.5)))
   (if (<= l 0.004) (exp (* M (- m))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 4.2e-300) {
		tmp = exp((n * (m * 0.5)));
	} else if (l <= 0.004) {
		tmp = exp((M * -m));
	} 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 <= 4.2d-300) then
        tmp = exp((n * (m * 0.5d0)))
    else if (l <= 0.004d0) then
        tmp = exp((m_1 * -m))
    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 <= 4.2e-300) {
		tmp = Math.exp((n * (m * 0.5)));
	} else if (l <= 0.004) {
		tmp = Math.exp((M * -m));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 4.2e-300:
		tmp = math.exp((n * (m * 0.5)))
	elif l <= 0.004:
		tmp = math.exp((M * -m))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 4.2e-300)
		tmp = exp(Float64(n * Float64(m * 0.5)));
	elseif (l <= 0.004)
		tmp = exp(Float64(M * Float64(-m)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 4.2e-300)
		tmp = exp((n * (m * 0.5)));
	elseif (l <= 0.004)
		tmp = exp((M * -m));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 4.2e-300], N[Exp[N[(n * N[(m * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 0.004], N[Exp[N[(M * (-m)), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.20000000000000007e-300

   1. Initial program 70.6%

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

      1. associate-/l* 70.6%

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

      2. associate--r- 70.6%

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

   3. Simplified 70.6%

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

      1. add-exp-log 68.9%

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

      2. *-commutative 68.9%

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

      3. log-prod 38.4%

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

   5. Applied egg-rr 11.8%

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

   6. Taylor expanded in K around 0 13.5%

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

      1. associate--l+ 13.5%

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

      2. +-commutative 13.5%

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

      3. +-commutative 13.5%

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

      4. cos-neg 13.5%

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

   8. Simplified 13.5%

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

   9. Taylor expanded in m around inf 24.3%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 24.3%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 24.3%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 24.3%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 24.3%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 24.3%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 24.3%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in n around inf 34.1%

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

       1. *-commutative 34.1%

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

       2. associate-*l* 34.1%

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

   14. Simplified 34.1%

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

   if 4.20000000000000007e-300 < l < 0.0040000000000000001

   1. Initial program 66.8%

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

      1. associate-/l* 66.6%

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

      2. associate--r- 66.6%

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

   3. Simplified 66.6%

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

      1. add-exp-log 66.5%

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

      2. *-commutative 66.5%

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

      3. log-prod 38.1%

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

   5. Applied egg-rr 10.4%

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

   6. Taylor expanded in K around 0 12.1%

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

      1. associate--l+ 12.1%

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

      2. +-commutative 12.1%

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

      3. +-commutative 12.1%

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

      4. cos-neg 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in m around inf 25.1%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 25.1%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 25.1%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 25.1%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 25.1%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 25.1%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 25.1%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in M around inf 37.6%

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

       1. associate-*r* 37.6%

          \[\leadsto e^{\color{blue}{\left(-1 \cdot m\right) \cdot M}} \]

       2. neg-mul-1 37.6%

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

   14. Simplified 37.6%

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

   if 0.0040000000000000001 < l

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in l around inf 71.0%

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

   5. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in M around 0 97.9%

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

      1. rec-exp 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-300}:\\ \;\;\;\;e^{n \cdot \left(m \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 0.004:\\ \;\;\;\;e^{M \cdot \left(-m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 10: 48.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;e^{M \cdot \left(-m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.41) (exp (* M (- m))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.41) {
		tmp = exp((M * -m));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 0.41d0) then
        tmp = exp((m_1 * -m))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.41) {
		tmp = Math.exp((M * -m));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.41:
		tmp = math.exp((M * -m))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.41)
		tmp = exp(Float64(M * Float64(-m)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.41)
		tmp = exp((M * -m));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.41], N[Exp[N[(M * (-m)), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.409999999999999976

   1. Initial program 69.1%

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

      1. associate-/l* 69.0%

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

      2. associate--r- 69.0%

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

   3. Simplified 69.0%

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

      1. add-exp-log 67.9%

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

      2. *-commutative 67.9%

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

      3. log-prod 38.3%

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

   5. Applied egg-rr 11.3%

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

   6. Taylor expanded in K around 0 12.9%

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

      1. associate--l+ 12.9%

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

      2. +-commutative 12.9%

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

      3. +-commutative 12.9%

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

      4. cos-neg 12.9%

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

   8. Simplified 12.9%

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

   9. Taylor expanded in m around inf 24.6%

      \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
   10. Step-by-step derivation

       1. fma-def 24.6%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

       2. unpow2 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

       3. *-commutative 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

       4. associate--l+ 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

       5. *-commutative 24.6%

          \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

   11. Simplified 24.6%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

   12. Taylor expanded in M around inf 30.4%

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

       1. associate-*r* 30.4%

          \[\leadsto e^{\color{blue}{\left(-1 \cdot m\right) \cdot M}} \]

       2. neg-mul-1 30.4%

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

   14. Simplified 30.4%

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

   if 0.409999999999999976 < l

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in l around inf 71.0%

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

   5. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in M around 0 97.9%

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

      1. rec-exp 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;e^{M \cdot \left(-m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 11: 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 69.8%

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

3. Simplified 69.8%

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

4. Taylor expanded in l around inf 28.6%

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

5. Taylor expanded in K around 0 37.0%

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

   1. cos-neg 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in M around 0 36.2%

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

   1. rec-exp 36.2%

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

10. Simplified 36.2%

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

11. Final simplification 36.2%

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

Alternative 12: 6.9% 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 69.8%

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

3. Simplified 69.8%

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

4. Taylor expanded in l around inf 28.6%

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

5. Taylor expanded in K around 0 37.0%

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

   1. cos-neg 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in l around 0 8.1%

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

9. Final simplification 8.1%

   \[\leadsto \cos M \]

Alternative 13: 6.9% accurate, 425.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. associate-/l* 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. associate--r- 69.7%

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

3. Simplified 69.7%

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

   1. add-exp-log 68.9%

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

   2. *-commutative 68.9%

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

   3. log-prod 40.4%

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

5. Applied egg-rr 11.4%

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

6. Taylor expanded in K around 0 14.3%

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

   1. associate--l+ 14.3%

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

   2. +-commutative 14.3%

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

   3. +-commutative 14.3%

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

   4. cos-neg 14.3%

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

8. Simplified 14.3%

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

9. Taylor expanded in m around inf 23.0%

   \[\leadsto e^{\color{blue}{0.25 \cdot {m}^{2} + \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m}} \]
10. Step-by-step derivation

    1. fma-def 23.0%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, {m}^{2}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)}} \]

    2. unpow2 23.0%

       \[\leadsto e^{\mathsf{fma}\left(0.25, \color{blue}{m \cdot m}, \left(\left(1 + 0.5 \cdot n\right) - M\right) \cdot m\right)} \]

    3. *-commutative 23.0%

       \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, \color{blue}{m \cdot \left(\left(1 + 0.5 \cdot n\right) - M\right)}\right)} \]

    4. associate--l+ 23.0%

       \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \color{blue}{\left(1 + \left(0.5 \cdot n - M\right)\right)}\right)} \]

    5. *-commutative 23.0%

       \[\leadsto e^{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(\color{blue}{n \cdot 0.5} - M\right)\right)\right)} \]

11. Simplified 23.0%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, m \cdot m, m \cdot \left(1 + \left(n \cdot 0.5 - M\right)\right)\right)}} \]

12. Taylor expanded in m around 0 8.1%

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

13. Final simplification 8.1%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023222 
(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)))))))