Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.4%

Time: 15.6s

Alternatives: 11

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

2. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

4. Simplified 97.1%

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

5. Final simplification 97.1%

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

Alternative 2: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;\cos \left(\frac{n \cdot K}{2} - M\right) \cdot e^{\left(\left(m - n\right) - {\left(n \cdot 0.5 - M\right)}^{2}\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 55.0)
   (*
    (cos (- (/ (* n K) 2.0) M))
    (exp (- (- (- m n) (pow (- (* n 0.5) M) 2.0)) l)))
   (* (cos M) (exp (* (* n n) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 55.0) {
		tmp = cos((((n * K) / 2.0) - M)) * exp((((m - n) - pow(((n * 0.5) - M), 2.0)) - l));
	} else {
		tmp = cos(M) * 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 <= 55.0d0) then
        tmp = cos((((n * k) / 2.0d0) - m_1)) * exp((((m - n) - (((n * 0.5d0) - m_1) ** 2.0d0)) - l))
    else
        tmp = cos(m_1) * 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 <= 55.0) {
		tmp = Math.cos((((n * K) / 2.0) - M)) * Math.exp((((m - n) - Math.pow(((n * 0.5) - M), 2.0)) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 55.0:
		tmp = math.cos((((n * K) / 2.0) - M)) * math.exp((((m - n) - math.pow(((n * 0.5) - M), 2.0)) - l))
	else:
		tmp = math.cos(M) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 55.0)
		tmp = Float64(cos(Float64(Float64(Float64(n * K) / 2.0) - M)) * exp(Float64(Float64(Float64(m - n) - (Float64(Float64(n * 0.5) - M) ^ 2.0)) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 55.0)
		tmp = cos((((n * K) / 2.0) - M)) * exp((((m - n) - (((n * 0.5) - M) ^ 2.0)) - l));
	else
		tmp = cos(M) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 55.0], N[(N[Cos[N[(N[(N[(n * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(m - n), $MachinePrecision] - N[Power[N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 55

   1. Initial program 77.2%

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

   2. Taylor expanded in m around 0 86.9%

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

   3. Taylor expanded in m around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. associate--r+ 68.5%

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

      3. unpow1 68.5%

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

      4. sqr-pow 47.0%

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

      5. fabs-sqr 47.0%

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

      6. sqr-pow 77.2%

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

      7. unpow1 77.2%

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

      8. *-commutative 77.2%

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

   5. Simplified 77.2%

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

   if 55 < n

   1. Initial program 66.7%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in n around inf 96.9%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

   7. Simplified 96.9%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;\cos \left(\frac{n \cdot K}{2} - M\right) \cdot e^{\left(\left(m - n\right) - {\left(n \cdot 0.5 - M\right)}^{2}\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 3: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;\cos \left(\frac{n \cdot K}{2} - M\right) \cdot e^{\left(\left(m - n\right) - M \cdot M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 54.0)
   (* (cos (- (/ (* n K) 2.0) M)) (exp (- (- (- m n) (* M M)) l)))
   (* (cos M) (exp (* (* n n) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 54.0) {
		tmp = cos((((n * K) / 2.0) - M)) * exp((((m - n) - (M * M)) - l));
	} else {
		tmp = cos(M) * 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 <= 54.0d0) then
        tmp = cos((((n * k) / 2.0d0) - m_1)) * exp((((m - n) - (m_1 * m_1)) - l))
    else
        tmp = cos(m_1) * 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 <= 54.0) {
		tmp = Math.cos((((n * K) / 2.0) - M)) * Math.exp((((m - n) - (M * M)) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 54.0:
		tmp = math.cos((((n * K) / 2.0) - M)) * math.exp((((m - n) - (M * M)) - l))
	else:
		tmp = math.cos(M) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 54.0)
		tmp = Float64(cos(Float64(Float64(Float64(n * K) / 2.0) - M)) * exp(Float64(Float64(Float64(m - n) - Float64(M * M)) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 54.0)
		tmp = cos((((n * K) / 2.0) - M)) * exp((((m - n) - (M * M)) - l));
	else
		tmp = cos(M) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 54.0], N[(N[Cos[N[(N[(N[(n * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(m - n), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 54

   1. Initial program 77.2%

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

   2. Taylor expanded in m around 0 86.9%

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

   3. Taylor expanded in m around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. associate--r+ 68.5%

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

      3. unpow1 68.5%

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

      4. sqr-pow 47.0%

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

      5. fabs-sqr 47.0%

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

      6. sqr-pow 77.2%

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

      7. unpow1 77.2%

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

      8. *-commutative 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in n around 0 65.5%

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

      1. unpow2 65.5%

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

   8. Simplified 65.5%

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

   if 54 < n

   1. Initial program 66.7%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in n around inf 96.9%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

   7. Simplified 96.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;\cos \left(\frac{n \cdot K}{2} - M\right) \cdot e^{\left(\left(m - n\right) - M \cdot M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 4: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;e^{\left(\left(m - n\right) - M \cdot M\right) - \ell} \cdot \cos \left(K \cdot \left(n \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 55.0)
   (* (exp (- (- (- m n) (* M M)) l)) (cos (* K (* n 0.5))))
   (* (cos M) (exp (* (* n n) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 55.0) {
		tmp = exp((((m - n) - (M * M)) - l)) * cos((K * (n * 0.5)));
	} else {
		tmp = cos(M) * 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 <= 55.0d0) then
        tmp = exp((((m - n) - (m_1 * m_1)) - l)) * cos((k * (n * 0.5d0)))
    else
        tmp = cos(m_1) * 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 <= 55.0) {
		tmp = Math.exp((((m - n) - (M * M)) - l)) * Math.cos((K * (n * 0.5)));
	} else {
		tmp = Math.cos(M) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 55.0:
		tmp = math.exp((((m - n) - (M * M)) - l)) * math.cos((K * (n * 0.5)))
	else:
		tmp = math.cos(M) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 55.0)
		tmp = Float64(exp(Float64(Float64(Float64(m - n) - Float64(M * M)) - l)) * cos(Float64(K * Float64(n * 0.5))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 55.0)
		tmp = exp((((m - n) - (M * M)) - l)) * cos((K * (n * 0.5)));
	else
		tmp = cos(M) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 55.0], N[(N[Exp[N[(N[(N[(m - n), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K * N[(n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 55

   1. Initial program 77.2%

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

   2. Taylor expanded in m around 0 86.9%

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

   3. Taylor expanded in m around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. associate--r+ 68.5%

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

      3. unpow1 68.5%

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

      4. sqr-pow 47.0%

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

      5. fabs-sqr 47.0%

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

      6. sqr-pow 77.2%

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

      7. unpow1 77.2%

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

      8. *-commutative 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in n around 0 65.5%

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

      1. unpow2 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in K around inf 64.9%

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

       1. *-commutative 64.9%

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

       2. associate-*l* 64.9%

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

   11. Simplified 64.9%

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

   if 55 < n

   1. Initial program 66.7%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in n around inf 96.9%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

   7. Simplified 96.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;e^{\left(\left(m - n\right) - M \cdot M\right) - \ell} \cdot \cos \left(K \cdot \left(n \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 5: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{if}\;m \leq -39:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-189}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* -0.25 (* m m))))))
   (if (<= m -39.0)
     t_0
     (if (<= m 3.4e-189)
       (* (cos M) (exp (* M (- M))))
       (if (<= m 1.6) (exp (- l)) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((-0.25 * (m * m)));
	double tmp;
	if (m <= -39.0) {
		tmp = t_0;
	} else if (m <= 3.4e-189) {
		tmp = cos(M) * exp((M * -M));
	} else if (m <= 1.6) {
		tmp = exp(-l);
	} 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 = cos(m_1) * exp(((-0.25d0) * (m * m)))
    if (m <= (-39.0d0)) then
        tmp = t_0
    else if (m <= 3.4d-189) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else if (m <= 1.6d0) then
        tmp = exp(-l)
    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.cos(M) * Math.exp((-0.25 * (m * m)));
	double tmp;
	if (m <= -39.0) {
		tmp = t_0;
	} else if (m <= 3.4e-189) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else if (m <= 1.6) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((-0.25 * (m * m)))
	tmp = 0
	if m <= -39.0:
		tmp = t_0
	elif m <= 3.4e-189:
		tmp = math.cos(M) * math.exp((M * -M))
	elif m <= 1.6:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))))
	tmp = 0.0
	if (m <= -39.0)
		tmp = t_0;
	elseif (m <= 3.4e-189)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	elseif (m <= 1.6)
		tmp = exp(Float64(-l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((-0.25 * (m * m)));
	tmp = 0.0;
	if (m <= -39.0)
		tmp = t_0;
	elseif (m <= 3.4e-189)
		tmp = cos(M) * exp((M * -M));
	elseif (m <= 1.6)
		tmp = exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -39.0], t$95$0, If[LessEqual[m, 3.4e-189], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.6], N[Exp[(-l)], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -39 or 1.6000000000000001 < m

   1. Initial program 66.2%

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

   2. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in m around inf 97.1%

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

      1. *-commutative 97.1%

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

      2. unpow2 97.1%

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

   7. Simplified 97.1%

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

   if -39 < m < 3.4000000000000001e-189

   1. Initial program 81.9%

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

   2. Taylor expanded in K around 0 96.5%

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

      1. cos-neg 96.5%

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

   4. Simplified 96.5%

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

   5. Taylor expanded in M around inf 60.4%

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

      1. mul-1-neg 60.4%

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

      2. unpow2 60.4%

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

      3. distribute-rgt-neg-in 60.4%

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

   7. Simplified 60.4%

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

   if 3.4000000000000001e-189 < m < 1.6000000000000001

   1. Initial program 87.1%

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

   2. Taylor expanded in l around inf 56.9%

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

      1. mul-1-neg 56.9%

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

   4. Simplified 56.9%

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

   5. Taylor expanded in K around 0 61.1%

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

      1. exp-neg 61.1%

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

      2. associate-*r/ 61.1%

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

      3. *-rgt-identity 61.1%

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

      4. cos-neg 61.1%

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

   7. Simplified 61.1%

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

   8. Taylor expanded in M around 0 61.1%

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

      1. rec-exp 61.1%

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

   10. Simplified 61.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -39:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-189}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \]

Alternative 6: 65.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -39:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-269}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -39.0)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= m 1.02e-269)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -39.0) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (m <= 1.02e-269) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * 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 (m <= (-39.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (m <= 1.02d-269) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * 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 (m <= -39.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (m <= 1.02e-269) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -39.0:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif m <= 1.02e-269:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -39.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= 1.02e-269)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -39.0)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (m <= 1.02e-269)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -39.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.02e-269], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -39

   1. Initial program 77.4%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in m around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

   7. Simplified 100.0%

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

   if -39 < m < 1.02000000000000002e-269

   1. Initial program 83.6%

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

   2. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in M around inf 60.3%

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

      1. mul-1-neg 60.3%

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

      2. unpow2 60.3%

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

      3. distribute-rgt-neg-in 60.3%

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

   7. Simplified 60.3%

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

   if 1.02000000000000002e-269 < m

   1. Initial program 69.2%

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

   2. Taylor expanded in K around 0 95.2%

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

      1. cos-neg 95.2%

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

   4. Simplified 95.2%

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

   5. Taylor expanded in n around inf 52.0%

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

      1. *-commutative 32.4%

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

      2. unpow2 32.4%

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

   7. Simplified 52.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -39:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-269}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 7: 72.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -7.5e-9)
   (* (cos M) (exp l))
   (if (<= l 720.0) (* (cos M) (exp (* M (- M)))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -7.5e-9) {
		tmp = cos(M) * exp(l);
	} else if (l <= 720.0) {
		tmp = cos(M) * 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 <= (-7.5d-9)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 720.0d0) then
        tmp = cos(m_1) * exp((m_1 * -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 <= -7.5e-9) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 720.0) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -7.5e-9:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 720.0:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -7.5e-9)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 720.0)
		tmp = Float64(cos(M) * 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 <= -7.5e-9)
		tmp = cos(M) * exp(l);
	elseif (l <= 720.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -7.5e-9], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 720.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.49999999999999933e-9

   1. Initial program 70.5%

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

   2. Taylor expanded in l around inf 20.5%

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

      1. mul-1-neg 20.5%

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

   4. Simplified 20.5%

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

   5. Taylor expanded in K around 0 20.9%

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

      1. exp-neg 20.9%

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

      2. associate-*r/ 20.9%

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

      3. *-rgt-identity 20.9%

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

      4. cos-neg 20.9%

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

   7. Simplified 20.9%

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

      1. expm1-log1p-u 17.2%

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

      2. expm1-udef 17.2%

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

      3. div-inv 17.2%

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

      4. exp-neg 17.2%

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

      5. add-sqr-sqrt 17.2%

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

      6. sqrt-unprod 17.2%

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

      7. sqr-neg 17.2%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. expm1-def 72.6%

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

       2. expm1-log1p 72.6%

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

       3. *-commutative 72.6%

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

   11. Simplified 72.6%

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

   if -7.49999999999999933e-9 < l < 720

   1. Initial program 76.2%

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

   2. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   4. Simplified 97.1%

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

   5. Taylor expanded in M around inf 59.7%

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

      1. mul-1-neg 59.7%

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

      2. unpow2 59.7%

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

      3. distribute-rgt-neg-in 59.7%

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

   7. Simplified 59.7%

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

   if 720 < l

   1. Initial program 75.3%

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

   2. Taylor expanded in l around inf 75.3%

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

      1. mul-1-neg 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0 100.0%

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

      1. rec-exp 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 8: 57.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -700:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-14}:\\ \;\;\;\;t_0 \cdot \left(\left(m \cdot m\right) \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= l -700.0)
     (* (cos M) (exp l))
     (if (<= l 2.5e-14) (* t_0 (* (* m m) (* K (* K -0.125)))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if (l <= -700.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 2.5e-14) {
		tmp = t_0 * ((m * m) * (K * (K * -0.125)));
	} 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(-l)
    if (l <= (-700.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 2.5d-14) then
        tmp = t_0 * ((m * m) * (k * (k * (-0.125d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-l);
	double tmp;
	if (l <= -700.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 2.5e-14) {
		tmp = t_0 * ((m * m) * (K * (K * -0.125)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if l <= -700.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 2.5e-14:
		tmp = t_0 * ((m * m) * (K * (K * -0.125)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -700.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 2.5e-14)
		tmp = Float64(t_0 * Float64(Float64(m * m) * Float64(K * Float64(K * -0.125))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if (l <= -700.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 2.5e-14)
		tmp = t_0 * ((m * m) * (K * (K * -0.125)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -700.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-14], N[(t$95$0 * N[(N[(m * m), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if l < -700

   1. Initial program 70.0%

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

   2. Taylor expanded in l around inf 20.8%

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

      1. mul-1-neg 20.8%

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

   4. Simplified 20.8%

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

   5. Taylor expanded in K around 0 21.1%

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

      1. exp-neg 21.1%

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

      2. associate-*r/ 21.1%

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

      3. *-rgt-identity 21.1%

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

      4. cos-neg 21.1%

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

   7. Simplified 21.1%

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

      1. expm1-log1p-u 17.5%

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

      2. expm1-udef 17.5%

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

      3. div-inv 17.5%

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

      4. exp-neg 17.5%

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

      5. add-sqr-sqrt 17.5%

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

      6. sqrt-unprod 17.5%

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

      7. sqr-neg 17.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 73.8%

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

   9. Applied egg-rr 73.8%

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

       1. expm1-def 73.8%

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

       2. expm1-log1p 73.8%

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

       3. *-commutative 73.8%

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

   11. Simplified 73.8%

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

   if -700 < l < 2.5000000000000001e-14

   1. Initial program 76.4%

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

   2. Taylor expanded in l around inf 13.7%

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

      1. mul-1-neg 13.7%

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

   4. Simplified 13.7%

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

   5. Taylor expanded in m around inf 13.5%

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

   6. Taylor expanded in K around 0 11.1%

      \[\leadsto \color{blue}{\left(1 + -0.125 \cdot \left({K}^{2} \cdot {m}^{2}\right)\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. unpow2 11.1%

         \[\leadsto \left(1 + -0.125 \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot {m}^{2}\right)\right) \cdot e^{-\ell} \]

      2. unpow2 11.1%

         \[\leadsto \left(1 + -0.125 \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{\left(m \cdot m\right)}\right)\right) \cdot e^{-\ell} \]

   8. Simplified 11.1%

      \[\leadsto \color{blue}{\left(1 + -0.125 \cdot \left(\left(K \cdot K\right) \cdot \left(m \cdot m\right)\right)\right)} \cdot e^{-\ell} \]

   9. Taylor expanded in K around inf 30.8%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left({K}^{2} \cdot {m}^{2}\right)\right)} \cdot e^{-\ell} \]
   10. Step-by-step derivation

       1. associate-*r* 30.8%

          \[\leadsto \color{blue}{\left(\left(-0.125 \cdot {K}^{2}\right) \cdot {m}^{2}\right)} \cdot e^{-\ell} \]

       2. unpow2 30.8%

          \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right) \cdot {m}^{2}\right) \cdot e^{-\ell} \]

       3. unpow2 30.8%

          \[\leadsto \left(\left(-0.125 \cdot \left(K \cdot K\right)\right) \cdot \color{blue}{\left(m \cdot m\right)}\right) \cdot e^{-\ell} \]

       4. associate-*r* 30.8%

          \[\leadsto \left(\color{blue}{\left(\left(-0.125 \cdot K\right) \cdot K\right)} \cdot \left(m \cdot m\right)\right) \cdot e^{-\ell} \]

   11. Simplified 30.8%

       \[\leadsto \color{blue}{\left(\left(\left(-0.125 \cdot K\right) \cdot K\right) \cdot \left(m \cdot m\right)\right)} \cdot e^{-\ell} \]

   if 2.5000000000000001e-14 < l

   1. Initial program 75.3%

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

   2. Taylor expanded in l around inf 72.8%

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

      1. mul-1-neg 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in K around 0 96.2%

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

      1. exp-neg 96.2%

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

      2. associate-*r/ 96.2%

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

      3. *-rgt-identity 96.2%

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

      4. cos-neg 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in M around 0 96.2%

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

      1. rec-exp 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -700:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-14}:\\ \;\;\;\;e^{-\ell} \cdot \left(\left(m \cdot m\right) \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 9: 43.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -700 \lor \neg \left(\ell \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(m \cdot m\right) \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (or (<= l -700.0) (not (<= l 2e-14)))
     t_0
     (* t_0 (* (* m m) (* K (* K -0.125)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double tmp;
	if ((l <= -700.0) || !(l <= 2e-14)) {
		tmp = t_0;
	} else {
		tmp = t_0 * ((m * m) * (K * (K * -0.125)));
	}
	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(-l)
    if ((l <= (-700.0d0)) .or. (.not. (l <= 2d-14))) then
        tmp = t_0
    else
        tmp = t_0 * ((m * m) * (k * (k * (-0.125d0))))
    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(-l);
	double tmp;
	if ((l <= -700.0) || !(l <= 2e-14)) {
		tmp = t_0;
	} else {
		tmp = t_0 * ((m * m) * (K * (K * -0.125)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	tmp = 0
	if (l <= -700.0) or not (l <= 2e-14):
		tmp = t_0
	else:
		tmp = t_0 * ((m * m) * (K * (K * -0.125)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if ((l <= -700.0) || !(l <= 2e-14))
		tmp = t_0;
	else
		tmp = Float64(t_0 * Float64(Float64(m * m) * Float64(K * Float64(K * -0.125))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	tmp = 0.0;
	if ((l <= -700.0) || ~((l <= 2e-14)))
		tmp = t_0;
	else
		tmp = t_0 * ((m * m) * (K * (K * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[Or[LessEqual[l, -700.0], N[Not[LessEqual[l, 2e-14]], $MachinePrecision]], t$95$0, N[(t$95$0 * N[(N[(m * m), $MachinePrecision] * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -700 or 2e-14 < l

   1. Initial program 73.0%

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

   2. Taylor expanded in l around inf 50.0%

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

      1. mul-1-neg 50.0%

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

   4. Simplified 50.0%

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

   5. Taylor expanded in K around 0 63.3%

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

      1. exp-neg 63.3%

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

      2. associate-*r/ 63.3%

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

      3. *-rgt-identity 63.3%

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

      4. cos-neg 63.3%

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

   7. Simplified 63.3%

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

   8. Taylor expanded in M around 0 62.6%

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

      1. rec-exp 62.6%

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

   10. Simplified 62.6%

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

   if -700 < l < 2e-14

   1. Initial program 76.4%

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

   2. Taylor expanded in l around inf 13.7%

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

      1. mul-1-neg 13.7%

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

   4. Simplified 13.7%

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

   5. Taylor expanded in m around inf 13.5%

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

   6. Taylor expanded in K around 0 11.1%

      \[\leadsto \color{blue}{\left(1 + -0.125 \cdot \left({K}^{2} \cdot {m}^{2}\right)\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. unpow2 11.1%

         \[\leadsto \left(1 + -0.125 \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot {m}^{2}\right)\right) \cdot e^{-\ell} \]

      2. unpow2 11.1%

         \[\leadsto \left(1 + -0.125 \cdot \left(\left(K \cdot K\right) \cdot \color{blue}{\left(m \cdot m\right)}\right)\right) \cdot e^{-\ell} \]

   8. Simplified 11.1%

      \[\leadsto \color{blue}{\left(1 + -0.125 \cdot \left(\left(K \cdot K\right) \cdot \left(m \cdot m\right)\right)\right)} \cdot e^{-\ell} \]

   9. Taylor expanded in K around inf 30.8%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left({K}^{2} \cdot {m}^{2}\right)\right)} \cdot e^{-\ell} \]
   10. Step-by-step derivation

       1. associate-*r* 30.8%

          \[\leadsto \color{blue}{\left(\left(-0.125 \cdot {K}^{2}\right) \cdot {m}^{2}\right)} \cdot e^{-\ell} \]

       2. unpow2 30.8%

          \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)}\right) \cdot {m}^{2}\right) \cdot e^{-\ell} \]

       3. unpow2 30.8%

          \[\leadsto \left(\left(-0.125 \cdot \left(K \cdot K\right)\right) \cdot \color{blue}{\left(m \cdot m\right)}\right) \cdot e^{-\ell} \]

       4. associate-*r* 30.8%

          \[\leadsto \left(\color{blue}{\left(\left(-0.125 \cdot K\right) \cdot K\right)} \cdot \left(m \cdot m\right)\right) \cdot e^{-\ell} \]

   11. Simplified 30.8%

       \[\leadsto \color{blue}{\left(\left(\left(-0.125 \cdot K\right) \cdot K\right) \cdot \left(m \cdot m\right)\right)} \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -700 \lor \neg \left(\ell \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell} \cdot \left(\left(m \cdot m\right) \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \end{array} \]

Alternative 10: 35.5% 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 74.6%

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

2. Taylor expanded in l around inf 33.2%

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

   1. mul-1-neg 33.2%

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

4. Simplified 33.2%

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

5. Taylor expanded in K around 0 40.4%

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

   1. exp-neg 40.4%

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

   2. associate-*r/ 40.4%

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

   3. *-rgt-identity 40.4%

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

   4. cos-neg 40.4%

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

7. Simplified 40.4%

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

8. Taylor expanded in M around 0 40.0%

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

   1. rec-exp 40.0%

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

10. Simplified 40.0%

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

11. Final simplification 40.0%

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

Alternative 11: 7.0% 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 74.6%

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

2. Taylor expanded in n around inf 38.2%

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

   1. *-commutative 38.2%

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

   2. unpow2 38.2%

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

4. Simplified 38.2%

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

5. Taylor expanded in n around 0 8.0%

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

6. Taylor expanded in K around 0 8.5%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \]
7. Step-by-step derivation

   1. cos-neg 8.5%

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

8. Simplified 8.5%

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

9. Final simplification 8.5%

   \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023271 
(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)))))))