Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.6% → 96.5%

Time: 20.1s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.6% 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

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

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 79.4%

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

3. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 87.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

5. Simplified 98.1%

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

6. Taylor expanded in n around 0 82.3%

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

   1. associate--r+ 82.3%

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

   2. unpow1 82.3%

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

   3. sqr-pow 40.9%

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

   4. fabs-sqr 40.9%

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

   5. sqr-pow 88.5%

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

   6. unpow1 88.5%

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

   7. *-commutative 88.5%

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

8. Simplified 88.5%

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

9. Final simplification 88.5%

   \[\leadsto \cos M \cdot e^{\left(\left(m - n\right) - \ell\right) - {\left(m \cdot 0.5 - M\right)}^{2}} \]
10. Add Preprocessing

Alternative 3: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {m}^{2}}\\ \mathbf{if}\;n \leq 2.7 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 720:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (pow m 2.0)))))
   (if (<= n 2.7e-198)
     t_0
     (if (<= n 1.6e-85)
       (/ (cos M) (exp l))
       (if (<= n 720.0) t_0 (* (cos M) (exp (- n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * pow(m, 2.0)));
	double tmp;
	if (n <= 2.7e-198) {
		tmp = t_0;
	} else if (n <= 1.6e-85) {
		tmp = cos(M) / exp(l);
	} else if (n <= 720.0) {
		tmp = t_0;
	} else {
		tmp = cos(M) * exp(-n);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (m ** 2.0d0)))
    if (n <= 2.7d-198) then
        tmp = t_0
    else if (n <= 1.6d-85) then
        tmp = cos(m_1) / exp(l)
    else if (n <= 720.0d0) then
        tmp = t_0
    else
        tmp = cos(m_1) * exp(-n)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * Math.pow(m, 2.0)));
	double tmp;
	if (n <= 2.7e-198) {
		tmp = t_0;
	} else if (n <= 1.6e-85) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (n <= 720.0) {
		tmp = t_0;
	} else {
		tmp = Math.cos(M) * Math.exp(-n);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * math.pow(m, 2.0)))
	tmp = 0
	if n <= 2.7e-198:
		tmp = t_0
	elif n <= 1.6e-85:
		tmp = math.cos(M) / math.exp(l)
	elif n <= 720.0:
		tmp = t_0
	else:
		tmp = math.cos(M) * math.exp(-n)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (m ^ 2.0)))
	tmp = 0.0
	if (n <= 2.7e-198)
		tmp = t_0;
	elseif (n <= 1.6e-85)
		tmp = Float64(cos(M) / exp(l));
	elseif (n <= 720.0)
		tmp = t_0;
	else
		tmp = Float64(cos(M) * exp(Float64(-n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (m ^ 2.0)));
	tmp = 0.0;
	if (n <= 2.7e-198)
		tmp = t_0;
	elseif (n <= 1.6e-85)
		tmp = cos(M) / exp(l);
	elseif (n <= 720.0)
		tmp = t_0;
	else
		tmp = cos(M) * exp(-n);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 2.7e-198], t$95$0, If[LessEqual[n, 1.6e-85], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 720.0], t$95$0, N[(N[Cos[M], $MachinePrecision] * N[Exp[(-n)], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.7000000000000002e-198 or 1.60000000000000014e-85 < n < 720

   1. Initial program 78.7%

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

   3. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in m around inf 58.0%

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

      1. *-commutative 58.0%

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

   8. Simplified 58.0%

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

   9. Taylor expanded in M around 0 58.0%

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

   if 2.7000000000000002e-198 < n < 1.60000000000000014e-85

   1. Initial program 85.8%

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

   3. Taylor expanded in l around inf 58.0%

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

      1. mul-1-neg 58.0%

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

   5. Simplified 58.0%

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

   6. Taylor expanded in K around 0 59.0%

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

      1. cos-neg 59.0%

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

      2. exp-neg 59.0%

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

      3. associate-*r/ 59.0%

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

      4. *-rgt-identity 59.0%

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

   8. Simplified 59.0%

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

   if 720 < n

   1. Initial program 78.9%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around 0 70.6%

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

      1. associate--r+ 70.6%

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

      2. unpow1 70.6%

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

      3. sqr-pow 14.0%

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

      4. fabs-sqr 14.0%

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

      5. sqr-pow 98.3%

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

      6. unpow1 98.3%

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

      7. *-commutative 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in n around inf 96.5%

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

       1. mul-1-neg 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.7 \cdot 10^{-198}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 720:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00013 \lor \neg \left(m \leq 55\right):\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -0.00013) (not (<= m 55.0)))
   (exp (* -0.25 (pow m 2.0)))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -0.00013) || !(m <= 55.0)) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m <= (-0.00013d0)) .or. (.not. (m <= 55.0d0))) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -0.00013) || !(m <= 55.0)) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -0.00013) or not (m <= 55.0):
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -0.00013) || !(m <= 55.0))
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -0.00013) || ~((m <= 55.0)))
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -0.00013], N[Not[LessEqual[m, 55.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.29999999999999989e-4 or 55 < m

   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. Add Preprocessing

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in m around inf 94.5%

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

      1. *-commutative 94.5%

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

   8. Simplified 94.5%

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

   9. Taylor expanded in M around 0 94.5%

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

   if -1.29999999999999989e-4 < m < 55

   1. Initial program 84.9%

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

   3. Taylor expanded in l around inf 40.3%

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

      1. mul-1-neg 40.3%

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

   5. Simplified 40.3%

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

   6. Taylor expanded in K around 0 45.8%

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

      1. cos-neg 45.8%

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

      2. exp-neg 45.8%

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

      3. associate-*r/ 45.8%

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

      4. *-rgt-identity 45.8%

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

   8. Simplified 45.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00013 \lor \neg \left(m \leq 55\right):\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+64}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) - {M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -5e+64)
   (exp (* -0.25 (pow m 2.0)))
   (exp (- (- (- m n) l) (pow M 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5e+64) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = exp((((m - n) - l) - pow(M, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-5d+64)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = exp((((m - n) - l) - (m_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5e+64) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.exp((((m - n) - l) - Math.pow(M, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -5e+64:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.exp((((m - n) - l) - math.pow(M, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -5e+64)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = exp(Float64(Float64(Float64(m - n) - l) - (M ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -5e+64)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = exp((((m - n) - l) - (M ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -5e+64], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5e64

   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. Add Preprocessing

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf 98.0%

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

      1. *-commutative 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in M around 0 98.0%

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

   if -5e64 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in n around 0 78.5%

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

      1. associate--r+ 78.5%

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

      2. unpow1 78.5%

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

      3. sqr-pow 47.9%

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

      4. fabs-sqr 47.9%

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

      5. sqr-pow 86.1%

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

      6. unpow1 86.1%

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

      7. *-commutative 86.1%

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

   8. Simplified 86.1%

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

   9. Taylor expanded in m around 0 71.8%

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

   10. Taylor expanded in M around 0 71.8%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left(m - n\right) - \ell\right) - {M}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+64}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(m - n\right) - \ell\right) - {M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in l around inf 31.1%

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

   1. mul-1-neg 31.1%

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

5. Simplified 31.1%

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

6. Taylor expanded in K around 0 36.9%

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

   1. cos-neg 36.9%

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

   2. exp-neg 36.9%

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

   3. associate-*r/ 36.9%

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

   4. *-rgt-identity 36.9%

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

8. Simplified 36.9%

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

9. Final simplification 36.9%

   \[\leadsto \frac{\cos M}{e^{\ell}} \]
10. Add Preprocessing

Alternative 7: 35.4% 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 79.4%

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

3. Taylor expanded in l around inf 31.1%

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

   1. mul-1-neg 31.1%

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

5. Simplified 31.1%

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

6. Taylor expanded in n around inf 32.8%

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

   1. *-commutative 32.8%

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

   2. *-commutative 32.8%

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

   3. associate-*l* 32.8%

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

8. Simplified 32.8%

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

9. Taylor expanded in n around 0 36.9%

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

10. Final simplification 36.9%

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

Alternative 8: 6.8% 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 79.4%

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

3. Taylor expanded in l around inf 31.1%

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

   1. mul-1-neg 31.1%

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

5. Simplified 31.1%

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

6. Taylor expanded in l around 0 8.4%

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

   1. *-commutative 8.4%

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

   2. *-commutative 8.4%

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

   3. associate-*l* 8.4%

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

8. Simplified 8.4%

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

9. Taylor expanded in K around 0 8.9%

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

    1. cos-neg 8.9%

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

11. Simplified 8.9%

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

12. Final simplification 8.9%

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

Alternative 9: 6.8% 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 79.4%

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

3. Taylor expanded in l around inf 31.1%

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

   1. mul-1-neg 31.1%

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

5. Simplified 31.1%

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

6. Taylor expanded in l around 0 8.4%

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

   1. *-commutative 8.4%

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

   2. *-commutative 8.4%

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

   3. associate-*l* 8.4%

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

8. Simplified 8.4%

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

9. Taylor expanded in K around 0 8.9%

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

    1. cos-neg 8.9%

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

11. Simplified 8.9%

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

12. Taylor expanded in M around 0 8.9%

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

13. Final simplification 8.9%

    \[\leadsto 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))))