Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.9%

Time: 20.7s

Alternatives: 7

Speedup: 3.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% 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 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. Add Preprocessing

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.02 \cdot 10^{+42} \lor \neg \left(M \leq 2.05 \cdot 10^{+46}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.02e+42) (not (<= M 2.05e+46)))
   (* (cos M) (exp (- (pow M 2.0))))
   (exp (- (- (fabs (- m n)) l) (* 0.25 (pow (+ m n) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.02e+42) || !(M <= 2.05e+46)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp(((fabs((m - n)) - l) - (0.25 * pow((m + n), 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_1 <= (-1.02d+42)) .or. (.not. (m_1 <= 2.05d+46))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((abs((m - n)) - l) - (0.25d0 * ((m + n) ** 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 <= -1.02e+42) || !(M <= 2.05e+46)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) - (0.25 * Math.pow((m + n), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.02e+42) or not (M <= 2.05e+46):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) - (0.25 * math.pow((m + n), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.02e+42) || !(M <= 2.05e+46))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(0.25 * (Float64(m + n) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.02e+42) || ~((M <= 2.05e+46)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp(((abs((m - n)) - l) - (0.25 * ((m + n) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.02e+42], N[Not[LessEqual[M, 2.05e+46]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.01999999999999996e42 or 2.05e46 < M

   1. Initial program 78.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 97.3%

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

      1. mul-1-neg 97.3%

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

   8. Simplified 97.3%

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

   if -1.01999999999999996e42 < M < 2.05e46

   1. Initial program 73.2%

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

   3. Taylor expanded in K around 0 94.3%

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

      1. cos-neg 94.3%

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

   5. Simplified 94.3%

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

   6. Taylor expanded in M around 0 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate--r+ 92.9%

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

      3. +-commutative 92.9%

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

   8. Simplified 92.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.02 \cdot 10^{+42} \lor \neg \left(M \leq 2.05 \cdot 10^{+46}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.05 \cdot 10^{+42} \lor \neg \left(M \leq 4.9 \cdot 10^{+45}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{0.25 \cdot \left(\left(m + n\right) \cdot \left(\left(-n\right) - m\right)\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.05e+42) (not (<= M 4.9e+45)))
   (* (cos M) (exp (- (pow M 2.0))))
   (exp (- (* 0.25 (* (+ m n) (- (- n) m))) l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.05e+42) || !(M <= 4.9e+45)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp(((0.25 * ((m + n) * (-n - m))) - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1.05d+42)) .or. (.not. (m_1 <= 4.9d+45))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((0.25d0 * ((m + n) * (-n - m))) - 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 <= -1.05e+42) || !(M <= 4.9e+45)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((0.25 * ((m + n) * (-n - m))) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.05e+42) or not (M <= 4.9e+45):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((0.25 * ((m + n) * (-n - m))) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1.05e+42) || !(M <= 4.9e+45))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(Float64(0.25 * Float64(Float64(m + n) * Float64(Float64(-n) - m))) - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.05e+42) || ~((M <= 4.9e+45)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp(((0.25 * ((m + n) * (-n - m))) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.05e+42], N[Not[LessEqual[M, 4.9e+45]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[((-n) - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.04999999999999998e42 or 4.9000000000000002e45 < M

   1. Initial program 78.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 97.3%

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

      1. mul-1-neg 97.3%

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

   8. Simplified 97.3%

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

   if -1.04999999999999998e42 < M < 4.9000000000000002e45

   1. Initial program 73.2%

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

   3. Taylor expanded in K around 0 94.3%

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

      1. cos-neg 94.3%

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

   5. Simplified 94.3%

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

   6. Taylor expanded in M around 0 92.9%

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

      1. fabs-sub 92.9%

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

      2. associate--r+ 92.9%

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

      3. +-commutative 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in l around inf 92.6%

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

       1. neg-mul-1 92.6%

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

   11. Simplified 92.6%

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

       1. unpow2 92.6%

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

   13. Applied egg-rr 92.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.05 \cdot 10^{+42} \lor \neg \left(M \leq 4.9 \cdot 10^{+45}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{0.25 \cdot \left(\left(m + n\right) \cdot \left(\left(-n\right) - m\right)\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ e^{0.25 \cdot \left(\left(m + n\right) \cdot \left(\left(-n\right) - m\right)\right) - \ell} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (* 0.25 (* (+ m n) (- (- n) m))) l)))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((0.25 * ((m + n) * (-n - m))) - 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(((0.25d0 * ((m + n) * (-n - m))) - l))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((0.25 * ((m + n) * (-n - m))) - l));
}

Python

def code(K, m, n, M, l):
	return math.exp(((0.25 * ((m + n) * (-n - m))) - l))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(0.25 * Float64(Float64(m + n) * Float64(Float64(-n) - m))) - l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((0.25 * ((m + n) * (-n - m))) - l));
end

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in M around 0 87.2%

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

   1. fabs-sub 87.2%

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

   2. associate--r+ 87.2%

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

   3. +-commutative 87.2%

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

8. Simplified 87.2%

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

9. Taylor expanded in l around inf 87.0%

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

    1. neg-mul-1 87.0%

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

11. Simplified 87.0%

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

    1. unpow2 87.0%

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

13. Applied egg-rr 87.0%

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

14. Final simplification 87.0%

    \[\leadsto e^{0.25 \cdot \left(\left(m + n\right) \cdot \left(\left(-n\right) - m\right)\right) - \ell} \]
15. Add Preprocessing

Alternative 5: 35.3% 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 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. Add Preprocessing

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in M around 0 87.2%

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

   1. fabs-sub 87.2%

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

   2. associate--r+ 87.2%

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

   3. +-commutative 87.2%

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

8. Simplified 87.2%

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

9. Taylor expanded in l around inf 87.0%

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

    1. neg-mul-1 87.0%

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

11. Simplified 87.0%

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

12. Taylor expanded in l around inf 39.4%

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

    1. neg-mul-1 39.4%

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

14. Simplified 39.4%

    \[\leadsto e^{\color{blue}{-\ell}} \]
15. Add Preprocessing

Alternative 6: 7.3% 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 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. Add Preprocessing

3. Taylor expanded in l around inf 34.8%

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

   1. mul-1-neg 34.8%

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

5. Simplified 34.8%

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

6. Taylor expanded in l around 0 6.5%

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

   1. neg-mul-1 6.5%

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

   2. unsub-neg 6.5%

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

8. Simplified 6.5%

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

9. Taylor expanded in K around 0 7.0%

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

    1. cos-neg 7.0%

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

11. Simplified 7.0%

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

12. Taylor expanded in l around 0 7.0%

    \[\leadsto \color{blue}{\cos M} \]
13. Add Preprocessing

Alternative 7: 7.1% accurate, 141.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(K, m, n, M, l)
	return Float64(1.0 - l)
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 - l), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in l around inf 34.8%

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

   1. mul-1-neg 34.8%

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

5. Simplified 34.8%

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

6. Taylor expanded in l around 0 6.5%

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

   1. neg-mul-1 6.5%

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

   2. unsub-neg 6.5%

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

8. Simplified 6.5%

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

9. Taylor expanded in K around 0 7.0%

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

    1. cos-neg 7.0%

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

11. Simplified 7.0%

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

12. Taylor expanded in M around 0 6.9%

    \[\leadsto \color{blue}{1 - \ell} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(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)))))))