Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.5%

Time: 14.4s

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: 76.6% 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 71.3%

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

3. Taylor expanded in K around 0 96.0%

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

   1. cos-neg 96.0%

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

5. Simplified 96.0%

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

6. Final simplification 96.0%

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

Alternative 2: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -27.5 \lor \neg \left(M \leq 31.5\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -27.5) (not (<= M 31.5)))
   (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 <= -27.5) || !(M <= 31.5)) {
		tmp = 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 <= (-27.5d0)) .or. (.not. (m_1 <= 31.5d0))) then
        tmp = 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 <= -27.5) || !(M <= 31.5)) {
		tmp = 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 <= -27.5) or not (M <= 31.5):
		tmp = 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 <= -27.5) || !(M <= 31.5))
		tmp = 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 <= -27.5) || ~((M <= 31.5)))
		tmp = 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, -27.5], N[Not[LessEqual[M, 31.5]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $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 < -27.5 or 31.5 < M

   1. Initial program 77.1%

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

   3. Taylor expanded in K around 0 99.1%

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

      1. cos-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in M around inf 98.2%

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

      1. mul-1-neg 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in M around 0 98.2%

      \[\leadsto \color{blue}{1} \cdot e^{-{M}^{2}} \]

   if -27.5 < M < 31.5

   1. Initial program 67.0%

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

   3. Taylor expanded in K around 0 93.7%

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

      1. cos-neg 93.7%

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

   5. Simplified 93.7%

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

   6. Taylor expanded in M around 0 93.7%

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

      1. fabs-sub 93.7%

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

      2. associate--r+ 93.7%

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

   8. Simplified 93.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27.5 \lor \neg \left(M \leq 31.5\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -7.2e-9)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (exp (- (- n m) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7.2e-9) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = exp(((n - m) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-7.2d-9)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = exp(((n - m) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7.2e-9) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.exp(((n - m) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -7.2e-9:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.exp(((n - m) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -7.2e-9)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = exp(Float64(Float64(n - m) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -7.2e-9)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = exp(((n - m) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -7.2e-9], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(n - m), $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -7.2e-9

   1. Initial program 68.6%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 100.0%

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

   if -7.2e-9 < m

   1. Initial program 72.3%

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

   3. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in M around 0 81.9%

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

      1. fabs-sub 81.9%

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

      2. associate--r+ 81.9%

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

   8. Simplified 81.9%

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

   9. Taylor expanded in m around 0 63.7%

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

       1. rem-square-sqrt 31.5%

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

       2. fabs-sqr 31.5%

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

       3. rem-square-sqrt 80.3%

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

   11. Simplified 80.3%

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

       1. unpow2 80.3%

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

   13. Applied egg-rr 80.3%

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

Alternative 4: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -7.2e-9)
   (exp (* -0.25 (pow m 2.0)))
   (exp (- (- n m) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7.2e-9) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = exp(((n - m) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-7.2d-9)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = exp(((n - m) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7.2e-9) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.exp(((n - m) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -7.2e-9:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.exp(((n - m) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -7.2e-9)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	else
		tmp = exp(Float64(Float64(n - m) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -7.2e-9)
		tmp = exp((-0.25 * (m ^ 2.0)));
	else
		tmp = exp(((n - m) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -7.2e-9], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n - m), $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -7.2e-9

   1. Initial program 68.6%

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

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 0 100.0%

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

      1. fabs-sub 100.0%

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

      2. associate--r+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -7.2e-9 < m

   1. Initial program 72.3%

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

   3. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in M around 0 81.9%

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

      1. fabs-sub 81.9%

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

      2. associate--r+ 81.9%

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

   8. Simplified 81.9%

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

   9. Taylor expanded in m around 0 63.7%

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

       1. rem-square-sqrt 31.5%

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

       2. fabs-sqr 31.5%

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

       3. rem-square-sqrt 80.3%

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

   11. Simplified 80.3%

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

       1. unpow2 80.3%

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

   13. Applied egg-rr 80.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-22} \lor \neg \left(n \leq 3.4 \cdot 10^{-13}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -6.2e-22) (not (<= n 3.4e-13)))
   (exp (* -0.25 (* n n)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -6.2e-22) || !(n <= 3.4e-13)) {
		tmp = exp((-0.25 * (n * n)));
	} 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 ((n <= (-6.2d-22)) .or. (.not. (n <= 3.4d-13))) then
        tmp = exp(((-0.25d0) * (n * n)))
    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 ((n <= -6.2e-22) || !(n <= 3.4e-13)) {
		tmp = Math.exp((-0.25 * (n * n)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -6.2e-22) or not (n <= 3.4e-13):
		tmp = math.exp((-0.25 * (n * n)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -6.2e-22) || !(n <= 3.4e-13))
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -6.2e-22) || ~((n <= 3.4e-13)))
		tmp = exp((-0.25 * (n * n)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -6.2e-22], N[Not[LessEqual[n, 3.4e-13]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.20000000000000025e-22 or 3.40000000000000015e-13 < n

   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. Add Preprocessing

   3. Taylor expanded in K around 0 97.8%

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

      1. cos-neg 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in M around 0 92.2%

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

      1. fabs-sub 92.2%

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

      2. associate--r+ 92.2%

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

   8. Simplified 92.2%

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

   9. Taylor expanded in n around inf 91.5%

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

       1. *-commutative 91.5%

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

   11. Simplified 91.5%

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

       1. unpow2 85.8%

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

   13. Applied egg-rr 91.5%

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

   if -6.20000000000000025e-22 < n < 3.40000000000000015e-13

   1. Initial program 77.3%

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

   3. Taylor expanded in K around 0 93.8%

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

      1. cos-neg 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in M around 0 80.4%

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

      1. fabs-sub 80.4%

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

      2. associate--r+ 80.4%

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

   8. Simplified 80.4%

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

   9. Taylor expanded in m around 0 32.2%

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

       1. rem-square-sqrt 15.5%

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

       2. fabs-sqr 15.5%

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

       3. rem-square-sqrt 51.9%

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

   11. Simplified 51.9%

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

   12. Taylor expanded in l around inf 45.0%

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

       1. mul-1-neg 45.0%

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

   14. Simplified 45.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-22} \lor \neg \left(n \leq 3.4 \cdot 10^{-13}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-102}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-m}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m 6e-102) (exp (- l)) (exp (- m))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= 6e-102) {
		tmp = exp(-l);
	} else {
		tmp = exp(-m);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= 6d-102) then
        tmp = exp(-l)
    else
        tmp = exp(-m)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= 6e-102) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp(-m);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= 6e-102:
		tmp = math.exp(-l)
	else:
		tmp = math.exp(-m)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= 6e-102)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= 6e-102)
		tmp = exp(-l);
	else
		tmp = exp(-m);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, 6e-102], N[Exp[(-l)], $MachinePrecision], N[Exp[(-m)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6e-102

   1. Initial program 72.6%

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

   3. Taylor expanded in K around 0 96.1%

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

      1. cos-neg 96.1%

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

   5. Simplified 96.1%

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

   6. Taylor expanded in M around 0 86.5%

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

      1. fabs-sub 86.5%

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

      2. associate--r+ 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in m around 0 62.3%

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

       1. rem-square-sqrt 40.6%

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

       2. fabs-sqr 40.6%

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

       3. rem-square-sqrt 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in l around inf 39.8%

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

       1. mul-1-neg 39.8%

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

   14. Simplified 39.8%

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

   if 6e-102 < m

   1. Initial program 68.8%

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

   3. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in M around 0 87.4%

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

      1. fabs-sub 87.4%

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

      2. associate--r+ 87.4%

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

   8. Simplified 87.4%

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

   9. Taylor expanded in m around 0 51.1%

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

       1. rem-square-sqrt 17.2%

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

       2. fabs-sqr 17.2%

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

       3. rem-square-sqrt 84.4%

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

   11. Simplified 84.4%

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

   12. Taylor expanded in m around inf 72.4%

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

       1. neg-mul-1 72.4%

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

   14. Simplified 72.4%

       \[\leadsto e^{\color{blue}{-m}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.2% 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 71.3%

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

3. Taylor expanded in K around 0 96.0%

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

   1. cos-neg 96.0%

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

5. Simplified 96.0%

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

6. Taylor expanded in M around 0 86.8%

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

   1. fabs-sub 86.8%

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

   2. associate--r+ 86.8%

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

8. Simplified 86.8%

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

9. Taylor expanded in m around 0 58.2%

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

    1. rem-square-sqrt 32.1%

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

    2. fabs-sqr 32.1%

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

    3. rem-square-sqrt 70.3%

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

11. Simplified 70.3%

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

12. Taylor expanded in l around inf 35.2%

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

    1. mul-1-neg 35.2%

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

14. Simplified 35.2%

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

Alternative 8: 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 71.3%

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

3. Taylor expanded in l around inf 28.0%

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

   1. mul-1-neg 28.0%

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

5. Simplified 28.0%

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

6. Taylor expanded in l around 0 6.0%

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

   1. *-commutative 6.0%

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

   2. *-commutative 6.0%

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

   3. associate-*l* 6.0%

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

   4. *-commutative 6.0%

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

8. Simplified 6.0%

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

9. Taylor expanded in K around 0 6.9%

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

    1. cos-neg 6.9%

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

11. Simplified 6.9%

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

Alternative 9: 7.0% 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 71.3%

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

3. Taylor expanded in K around 0 96.0%

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

   1. cos-neg 96.0%

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

5. Simplified 96.0%

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

6. Taylor expanded in M around 0 86.8%

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

   1. fabs-sub 86.8%

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

   2. associate--r+ 86.8%

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

8. Simplified 86.8%

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

9. Taylor expanded in n around inf 54.6%

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

    1. *-commutative 54.6%

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

11. Simplified 54.6%

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

12. Taylor expanded in n around 0 6.9%

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

Reproduce

herbie shell --seed 2024149 
(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)))))))