Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.8% → 96.6%

Time: 17.3s

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: 76.8% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- n m)) 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((n - m)) - 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((n - m)) - 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((n - m)) - 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((n - m)) - 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(n - m)) - 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((n - m)) - 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[(n - m), $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 78.0%

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

   1. associate-/l* 77.9%

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

   2. +-commutative 77.9%

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

   3. fabs-sub 77.9%

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

   4. +-commutative 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in K around 0 96.2%

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

   1. cos-neg 96.2%

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

7. Simplified 96.2%

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

8. Final simplification 96.2%

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

Alternative 2: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1400000 \lor \neg \left(M \leq 2.1 \cdot 10^{+16}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1400000.0) (not (<= M 2.1e+16)))
   (* (cos M) (exp (- (pow M 2.0))))
   (exp (- (fabs (- n m)) (+ 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 <= -1400000.0) || !(M <= 2.1e+16)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp((fabs((n - m)) - (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 <= (-1400000.0d0)) .or. (.not. (m_1 <= 2.1d+16))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp((abs((n - m)) - (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 <= -1400000.0) || !(M <= 2.1e+16)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((Math.abs((n - m)) - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1400000.0) or not (M <= 2.1e+16):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((math.fabs((n - m)) - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1400000.0) || !(M <= 2.1e+16))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(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 <= -1400000.0) || ~((M <= 2.1e+16)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp((abs((n - m)) - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1400000.0], N[Not[LessEqual[M, 2.1e+16]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.4e6 or 2.1e16 < M

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. +-commutative 76.6%

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

      3. fabs-sub 76.6%

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

      4. +-commutative 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in M around inf 96.9%

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

      1. mul-1-neg 96.9%

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

   10. Simplified 96.9%

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

   if -1.4e6 < M < 2.1e16

   1. Initial program 79.5%

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

      1. associate-/l* 79.2%

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

      2. +-commutative 79.2%

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

      3. fabs-sub 79.2%

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

      4. +-commutative 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0 88.6%

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

      1. cos-neg 88.6%

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

      2. associate-*r* 88.6%

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

      3. sin-neg 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in M around 0 94.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1400000 \lor \neg \left(M \leq 2.1 \cdot 10^{+16}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1000000 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - \left(m - \ell\right)\right) + {\left(m + n\right)}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1000000.0) (not (<= M 27.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (exp (+ (- n (- m l)) (* (pow (+ m n) 2.0) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1000000.0) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp(((n - (m - l)) + (pow((m + n), 2.0) * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-1000000.0d0)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((n - (m - l)) + (((m + n) ** 2.0d0) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1000000.0) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((n - (m - l)) + (Math.pow((m + n), 2.0) * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1000000.0) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((n - (m - l)) + (math.pow((m + n), 2.0) * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -1000000.0) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(Float64(n - Float64(m - l)) + Float64((Float64(m + n) ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1000000.0) || ~((M <= 27.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp(((n - (m - l)) + (((m + n) ^ 2.0) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1000000.0], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(n - N[(m - l), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1e6 or 27 < M

   1. Initial program 76.7%

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

      1. associate-/l* 76.7%

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

      2. +-commutative 76.7%

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

      3. fabs-sub 76.7%

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

      4. +-commutative 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in M around inf 96.3%

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

      1. mul-1-neg 96.3%

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

   10. Simplified 96.3%

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

   if -1e6 < M < 27

   1. Initial program 79.5%

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

      1. associate-/l* 79.2%

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

      2. +-commutative 79.2%

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

      3. fabs-sub 79.2%

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

      4. +-commutative 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0 88.9%

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

      1. cos-neg 88.9%

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

      2. associate-*r* 88.9%

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

      3. sin-neg 88.9%

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

   7. Simplified 88.9%

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

   8. Taylor expanded in M around 0 93.8%

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

      1. associate--r+ 93.8%

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

      2. sub-neg 93.8%

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

      3. sub-neg 93.8%

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

      4. add-sqr-sqrt 48.8%

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

      5. fabs-sqr 48.8%

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

      6. add-sqr-sqrt 93.8%

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

      7. add-sqr-sqrt 48.9%

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

      8. sqrt-unprod 70.4%

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

      9. sqr-neg 70.4%

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

      10. sqrt-unprod 36.9%

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

      11. add-sqr-sqrt 82.3%

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

      12. +-commutative 82.3%

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

   10. Applied egg-rr 82.3%

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

       1. associate-+l- 82.3%

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

       2. distribute-lft-neg-in 82.3%

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

       3. metadata-eval 82.3%

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

   12. Simplified 82.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1000000 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - \left(m - \ell\right)\right) + {\left(m + n\right)}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left(n - \left(m - \ell\right)\right) + {\left(m + n\right)}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 740.0)
   (exp (+ (- n (- m l)) (* (pow (+ m n) 2.0) -0.25)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 740.0) {
		tmp = exp(((n - (m - l)) + (pow((m + n), 2.0) * -0.25)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 740.0d0) then
        tmp = exp(((n - (m - l)) + (((m + n) ** 2.0d0) * (-0.25d0))))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 740.0) {
		tmp = Math.exp(((n - (m - l)) + (Math.pow((m + n), 2.0) * -0.25)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 740.0:
		tmp = math.exp(((n - (m - l)) + (math.pow((m + n), 2.0) * -0.25)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 740.0)
		tmp = exp(Float64(Float64(n - Float64(m - l)) + Float64((Float64(m + n) ^ 2.0) * -0.25)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 740.0)
		tmp = exp(((n - (m - l)) + (((m + n) ^ 2.0) * -0.25)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 740.0], N[Exp[N[(N[(n - N[(m - l), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 740

   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. Step-by-step derivation

      1. associate-/l* 76.9%

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

      2. +-commutative 76.9%

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

      3. fabs-sub 76.9%

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

      4. +-commutative 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in K around 0 82.6%

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

      1. cos-neg 82.6%

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

      2. associate-*r* 82.6%

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

      3. sin-neg 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in M around 0 81.9%

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

      1. associate--r+ 81.9%

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

      2. sub-neg 81.9%

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

      3. sub-neg 81.9%

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

      4. add-sqr-sqrt 37.0%

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

      5. fabs-sqr 37.0%

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

      6. add-sqr-sqrt 81.9%

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

      7. add-sqr-sqrt 55.8%

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

      8. sqrt-unprod 71.5%

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

      9. sqr-neg 71.5%

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

      10. sqrt-unprod 26.1%

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

      11. add-sqr-sqrt 83.3%

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

      12. +-commutative 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. associate-+l- 83.3%

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

       2. distribute-lft-neg-in 83.3%

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

       3. metadata-eval 83.3%

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

   12. Simplified 83.3%

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

   if 740 < l

   1. Initial program 81.0%

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

      1. associate-/l* 81.0%

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

      2. +-commutative 81.0%

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

      3. fabs-sub 81.0%

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

      4. +-commutative 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in K around 0 84.1%

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

      1. cos-neg 84.1%

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

      2. associate-*r* 84.1%

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

      3. sin-neg 84.1%

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

   7. Simplified 84.1%

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

   8. Taylor expanded in M around 0 100.0%

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

   9. Taylor expanded in l around inf 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left(n - \left(m - \ell\right)\right) + {\left(m + n\right)}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.0072:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -3.5e-9)
   (exp l)
   (if (<= l 0.0072) (exp (* -0.25 (pow n 2.0))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -3.5e-9) {
		tmp = exp(l);
	} else if (l <= 0.0072) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-3.5d-9)) then
        tmp = exp(l)
    else if (l <= 0.0072d0) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -3.5e-9) {
		tmp = Math.exp(l);
	} else if (l <= 0.0072) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -3.5e-9:
		tmp = math.exp(l)
	elif l <= 0.0072:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -3.5e-9)
		tmp = exp(l);
	elseif (l <= 0.0072)
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -3.5e-9)
		tmp = exp(l);
	elseif (l <= 0.0072)
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -3.5e-9], N[Exp[l], $MachinePrecision], If[LessEqual[l, 0.0072], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999999e-9

   1. Initial program 63.5%

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

      1. associate-/l* 63.5%

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

      2. +-commutative 63.5%

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

      3. fabs-sub 63.5%

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

      4. +-commutative 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in K around 0 71.4%

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

      1. cos-neg 71.4%

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

      2. associate-*r* 71.4%

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

      3. sin-neg 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in M around 0 73.3%

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

      1. associate--r+ 73.3%

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

      2. sub-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. add-sqr-sqrt 33.4%

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

      5. fabs-sqr 33.4%

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

      6. add-sqr-sqrt 73.3%

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

      7. add-sqr-sqrt 73.3%

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

      8. sqrt-unprod 41.6%

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

      9. sqr-neg 41.6%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 78.2%

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

      12. +-commutative 78.2%

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

   10. Applied egg-rr 78.2%

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

       1. associate-+l- 78.2%

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

       2. distribute-lft-neg-in 78.2%

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

       3. metadata-eval 78.2%

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

   12. Simplified 78.2%

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

   13. Taylor expanded in l around inf 72.0%

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

   if -3.4999999999999999e-9 < l < 0.0071999999999999998

   1. Initial program 84.0%

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

      1. associate-/l* 83.8%

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

      2. +-commutative 83.8%

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

      3. fabs-sub 83.8%

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

      4. +-commutative 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in K around 0 88.5%

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

      1. cos-neg 88.5%

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

      2. associate-*r* 88.5%

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

      3. sin-neg 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in M around 0 86.4%

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

   9. Taylor expanded in n around inf 61.2%

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

   if 0.0071999999999999998 < l

   1. Initial program 80.3%

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

      1. associate-/l* 80.3%

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

      2. +-commutative 80.3%

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

      3. fabs-sub 80.3%

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

      4. +-commutative 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in K around 0 83.3%

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

      1. cos-neg 83.3%

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

      2. associate-*r* 83.3%

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

      3. sin-neg 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in M around 0 98.7%

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

   9. Taylor expanded in l around inf 97.2%

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

       1. neg-mul-1 97.2%

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

   11. Simplified 97.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.0072:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if n < 55

   1. Initial program 79.7%

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

      1. associate-/l* 79.5%

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

      2. +-commutative 79.5%

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

      3. fabs-sub 79.5%

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

      4. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in K around 0 84.5%

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

      1. cos-neg 84.5%

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

      2. associate-*r* 84.5%

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

      3. sin-neg 84.5%

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

   7. Simplified 84.5%

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

   8. Taylor expanded in M around 0 83.8%

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

   9. Taylor expanded in m around inf 55.8%

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

       1. *-commutative 55.8%

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

   11. Simplified 55.8%

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

   if 55 < n

   1. Initial program 72.4%

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

      1. associate-/l* 72.4%

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

      2. +-commutative 72.4%

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

      3. fabs-sub 72.4%

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

      4. +-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in K around 0 77.6%

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

      1. cos-neg 77.6%

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

      2. associate-*r* 77.6%

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

      3. sin-neg 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in M around 0 94.9%

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

   9. Taylor expanded in n around inf 94.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 55:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{-9}:\\ \;\;\;\;e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -1.16e-9) (exp l) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1.16e-9) {
		tmp = exp(l);
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-1.16d-9)) then
        tmp = exp(l)
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1.16e-9) {
		tmp = Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -1.16e-9:
		tmp = math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -1.16e-9)
		tmp = exp(l);
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -1.16e-9)
		tmp = exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -1.16e-9], N[Exp[l], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.15999999999999992e-9

   1. Initial program 64.1%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fabs-sub 64.1%

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

      4. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in K around 0 71.9%

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

      1. cos-neg 71.9%

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

      2. associate-*r* 71.9%

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

      3. sin-neg 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in M around 0 73.7%

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

      1. associate--r+ 73.7%

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

      2. sub-neg 73.7%

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

      3. sub-neg 73.7%

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

      4. add-sqr-sqrt 32.9%

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

      5. fabs-sqr 32.9%

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

      6. add-sqr-sqrt 73.7%

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

      7. add-sqr-sqrt 73.7%

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

      8. sqrt-unprod 42.5%

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

      9. sqr-neg 42.5%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 78.5%

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

      12. +-commutative 78.5%

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

   10. Applied egg-rr 78.5%

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

       1. associate-+l- 78.5%

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

       2. distribute-lft-neg-in 78.5%

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

       3. metadata-eval 78.5%

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

   12. Simplified 78.5%

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

   13. Taylor expanded in l around inf 70.9%

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

   if -1.15999999999999992e-9 < l

   1. Initial program 82.7%

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

      1. associate-/l* 82.5%

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

      2. +-commutative 82.5%

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

      3. fabs-sub 82.5%

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

      4. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in K around 0 86.7%

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

      1. cos-neg 86.7%

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

      2. associate-*r* 86.7%

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

      3. sin-neg 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in M around 0 90.5%

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

   9. Taylor expanded in l around inf 39.6%

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

       1. neg-mul-1 39.6%

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

   11. Simplified 39.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{-9}:\\ \;\;\;\;e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 7.2% 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 78.0%

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

   1. associate-/l* 77.9%

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

   2. +-commutative 77.9%

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

   3. fabs-sub 77.9%

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

   4. +-commutative 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in K around 0 96.2%

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

   1. cos-neg 96.2%

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

7. Simplified 96.2%

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

8. Taylor expanded in m around inf 54.1%

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

   1. *-commutative 54.1%

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

10. Simplified 54.1%

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

11. Taylor expanded in m around 0 5.9%

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

    1. +-commutative 5.9%

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

    2. *-commutative 5.9%

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

13. Simplified 5.9%

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

14. Taylor expanded in m around 0 6.4%

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

15. Final simplification 6.4%

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

Alternative 9: 24.7% 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(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 78.0%

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

   1. associate-/l* 77.9%

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

   2. +-commutative 77.9%

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

   3. fabs-sub 77.9%

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

   4. +-commutative 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in K around 0 83.0%

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

   1. cos-neg 83.0%

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

   2. associate-*r* 83.0%

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

   3. sin-neg 83.0%

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

7. Simplified 83.0%

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

8. Taylor expanded in M around 0 86.3%

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

   1. associate--r+ 86.3%

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

   2. sub-neg 86.3%

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

   3. sub-neg 86.3%

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

   4. add-sqr-sqrt 39.6%

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

   5. fabs-sqr 39.6%

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

   6. add-sqr-sqrt 86.3%

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

   7. add-sqr-sqrt 42.1%

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

   8. sqrt-unprod 60.4%

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

   9. sqr-neg 60.4%

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

   10. sqrt-unprod 35.8%

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

   11. add-sqr-sqrt 78.9%

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

   12. +-commutative 78.9%

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

10. Applied egg-rr 78.9%

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

    1. associate-+l- 78.9%

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

    2. distribute-lft-neg-in 78.9%

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

    3. metadata-eval 78.9%

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

12. Simplified 78.9%

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

13. Taylor expanded in l around inf 23.0%

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

14. Final simplification 23.0%

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

Reproduce

herbie shell --seed 2024024 
(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)))))))