Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.8%

Time: 24.2s

Alternatives: 14

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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: 96.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos M)
  (pow E (- (- (fabs (- m n)) l) (pow (fma 0.5 (+ m n) (- M)) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * pow(((double) M_E), ((fabs((m - n)) - l) - pow(fma(0.5, (m + n), -M), 2.0)));
}

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * (exp(1) ^ Float64(Float64(abs(Float64(m - n)) - l) - (fma(0.5, Float64(m + n), Float64(-M)) ^ 2.0))))
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-/l* 76.5%

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

   2. +-commutative 76.5%

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

   3. fabs-sub 76.5%

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

   4. +-commutative 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in K around 0 97.0%

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

   1. cos-neg 97.0%

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

7. Simplified 97.0%

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

   1. *-un-lft-identity 97.0%

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

   2. exp-prod 97.0%

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

   3. associate--r- 97.0%

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

   4. div-inv 97.0%

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

   5. metadata-eval 97.0%

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

   6. fma-neg 97.0%

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

9. Applied egg-rr 97.0%

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

11. Simplified 97.0%

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

12. Final simplification 97.0%

    \[\leadsto \cos M \cdot {e}^{\left(\left(\left|m - n\right| - \ell\right) - {\left(\mathsf{fma}\left(0.5, m + n, -M\right)\right)}^{2}\right)} \]
13. Add Preprocessing

Alternative 2: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-/l* 76.5%

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

   2. +-commutative 76.5%

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

   3. fabs-sub 76.5%

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

   4. +-commutative 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in K around 0 97.0%

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

   1. cos-neg 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 3: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1000000000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot {e}^{\left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell - \left|m - n\right|\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1000000000.0)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (*
    (cos M)
    (pow
     E
     (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- l (fabs (- m n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1000000000.0) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * pow(((double) M_E), ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - fabs((m - n)))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1000000000.0) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.pow(Math.E, ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - Math.abs((m - n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1000000000.0:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.pow(math.e, ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - math.fabs((m - n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1000000000.0)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	else
		tmp = Float64(cos(M) * (exp(1) ^ Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - Float64(l - abs(Float64(m - n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1000000000.0)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * (2.71828182845904523536 ^ ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - abs((m - n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Power[E, N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1e9

   1. Initial program 68.9%

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

      1. associate-/l* 68.9%

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

      2. +-commutative 68.9%

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

      3. fabs-sub 68.9%

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

      4. +-commutative 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. unpow2 91.9%

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

      3. distribute-rgt-out 95.2%

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

      4. *-commutative 95.2%

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

      5. *-commutative 95.2%

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

   10. Simplified 95.2%

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

   11. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if -1e9 < m

   1. Initial program 77.9%

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

      1. associate-/l* 79.0%

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

      2. +-commutative 79.0%

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

      3. fabs-sub 79.0%

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

      4. +-commutative 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in K around 0 96.0%

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

      1. cos-neg 96.0%

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

   7. Simplified 96.0%

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

      1. *-un-lft-identity 96.0%

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

      2. exp-prod 96.0%

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

      3. associate--r- 96.0%

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

      4. div-inv 96.0%

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

      5. metadata-eval 96.0%

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

      6. fma-neg 96.0%

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

   9. Applied egg-rr 96.0%

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

   11. Simplified 96.0%

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

   12. Taylor expanded in m around 0 80.3%

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

       1. unpow2 80.3%

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

       2. distribute-rgt-in 82.4%

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

   14. Simplified 82.4%

       \[\leadsto \cos M \cdot {e}^{\left(\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot n - M\right) \cdot \left(m + \left(0.5 \cdot n - M\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1000000000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot {e}^{\left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell - \left|m - n\right|\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{+24}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1e+24)
   (*
    (cos M)
    (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- (fabs (- m n)) l))))
   (* (cos M) (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e+24) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (fabs((m - n)) - l)));
	} else {
		tmp = cos(M) * 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 <= 1d+24) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (abs((m - n)) - l)))
    else
        tmp = cos(m_1) * 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 <= 1e+24) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (Math.abs((m - n)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1e+24:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (math.fabs((m - n)) - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1e+24)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(abs(Float64(m - n)) - l))));
	else
		tmp = Float64(cos(M) * 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 <= 1e+24)
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (abs((m - n)) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1e+24], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 9.9999999999999998e23

   1. Initial program 74.4%

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

      1. associate-/l* 75.4%

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

      2. +-commutative 75.4%

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

      3. fabs-sub 75.4%

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

      4. +-commutative 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in n around 0 77.8%

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

      1. +-commutative 77.8%

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

      2. unpow2 77.8%

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

      3. distribute-rgt-out 83.3%

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

      4. *-commutative 83.3%

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

      5. *-commutative 83.3%

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

   10. Simplified 83.3%

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

   if 9.9999999999999998e23 < n

   1. Initial program 81.1%

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

      1. associate-/l* 81.1%

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

      2. +-commutative 81.1%

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

      3. fabs-sub 81.1%

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

      4. +-commutative 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in l around 0 98.1%

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

   9. Taylor expanded in n around inf 98.1%

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

       1. *-commutative 98.1%

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

   11. Simplified 98.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{+24}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -10000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell - \left|m - n\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -10000.0)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (*
    (cos M)
    (exp (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- l (fabs (- m n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -10000.0) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - fabs((m - 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 <= (-10000.0d0)) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - (l - abs((m - 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 <= -10000.0) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - Math.abs((m - n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -10000.0:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - math.fabs((m - n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -10000.0)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - Float64(l - abs(Float64(m - n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -10000.0)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l - abs((m - n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -10000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1e4

   1. Initial program 68.9%

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

      1. associate-/l* 68.9%

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

      2. +-commutative 68.9%

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

      3. fabs-sub 68.9%

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

      4. +-commutative 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. unpow2 91.9%

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

      3. distribute-rgt-out 95.2%

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

      4. *-commutative 95.2%

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

      5. *-commutative 95.2%

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

   10. Simplified 95.2%

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

   11. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if -1e4 < m

   1. Initial program 77.9%

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

      1. associate-/l* 79.0%

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

      2. +-commutative 79.0%

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

      3. fabs-sub 79.0%

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

      4. +-commutative 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in K around 0 96.0%

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

      1. cos-neg 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in m around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. unpow2 80.3%

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

      3. distribute-rgt-out 82.4%

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

      4. *-commutative 82.4%

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

      5. *-commutative 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right)}\right) - \left(\ell - \left|n - m\right|\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -10000:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell - \left|m - n\right|\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.25 \cdot 10^{-46}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -2.25e-46)
   (* (cos M) (exp (* M (- n M))))
   (if (<= M 1.1e-13)
     (* (cos M) (exp (* (pow m 2.0) -0.25)))
     (* (cos M) (exp (- (pow M 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -2.25e-46) {
		tmp = cos(M) * exp((M * (n - M)));
	} else if (M <= 1.1e-13) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * exp(-pow(M, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= (-2.25d-46)) then
        tmp = cos(m_1) * exp((m_1 * (n - m_1)))
    else if (m_1 <= 1.1d-13) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -2.25e-46) {
		tmp = Math.cos(M) * Math.exp((M * (n - M)));
	} else if (M <= 1.1e-13) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -2.25e-46:
		tmp = math.cos(M) * math.exp((M * (n - M)))
	elif M <= 1.1e-13:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -2.25e-46)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(n - M))));
	elseif (M <= 1.1e-13)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	else
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -2.25e-46)
		tmp = cos(M) * exp((M * (n - M)));
	elseif (M <= 1.1e-13)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -2.25e-46], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.1e-13], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.25e-46

   1. Initial program 76.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 98.7%

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

      1. cos-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in n around 0 78.3%

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

      1. +-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. distribute-rgt-out 88.6%

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

      4. *-commutative 88.6%

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

      5. *-commutative 88.6%

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

   10. Simplified 88.6%

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

   11. Taylor expanded in m around 0 79.8%

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

       1. associate--r+ 79.8%

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

       2. associate-*r* 79.8%

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

       3. neg-mul-1 79.8%

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

       4. cancel-sign-sub 79.8%

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

   13. Simplified 79.8%

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

   14. Taylor expanded in M around inf 77.2%

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

       1. +-commutative 77.2%

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

       2. mul-1-neg 77.2%

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

       3. unpow2 77.2%

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

       4. distribute-rgt-neg-in 77.2%

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

       5. distribute-lft-in 84.9%

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

       6. sub-neg 84.9%

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

   16. Simplified 84.9%

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

   if -2.25e-46 < M < 1.09999999999999998e-13

   1. Initial program 73.1%

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

      1. associate-/l* 75.0%

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

      2. +-commutative 75.0%

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

      3. fabs-sub 75.0%

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

      4. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in K around 0 93.7%

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

      1. cos-neg 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in n around 0 64.8%

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

      1. +-commutative 64.8%

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

      2. unpow2 64.8%

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

      3. distribute-rgt-out 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

   10. Simplified 67.6%

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

   11. Taylor expanded in m around inf 59.0%

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

       1. *-commutative 59.0%

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

   13. Simplified 59.0%

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

   if 1.09999999999999998e-13 < M

   1. Initial program 78.6%

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

      1. associate-/l* 78.6%

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

      2. +-commutative 78.6%

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

      3. fabs-sub 78.6%

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

      4. +-commutative 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 78.7%

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

      1. +-commutative 78.7%

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

      2. unpow2 78.7%

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

      3. distribute-rgt-out 88.8%

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

      4. *-commutative 88.8%

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

      5. *-commutative 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in M around inf 95.8%

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

       1. mul-1-neg 95.8%

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

   13. Simplified 95.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.25 \cdot 10^{-46}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.5 \cdot 10^{-293}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 56:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 8.5e-293)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (if (<= n 56.0)
     (* (cos M) (exp (+ (- n m) (- (* M (- n M)) l))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 8.5e-293) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else if (n <= 56.0) {
		tmp = cos(M) * exp(((n - m) + ((M * (n - M)) - l)));
	} else {
		tmp = cos(M) * 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 <= 8.5d-293) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else if (n <= 56.0d0) then
        tmp = cos(m_1) * exp(((n - m) + ((m_1 * (n - m_1)) - l)))
    else
        tmp = cos(m_1) * 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 <= 8.5e-293) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (n <= 56.0) {
		tmp = Math.cos(M) * Math.exp(((n - m) + ((M * (n - M)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 8.5e-293:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	elif n <= 56.0:
		tmp = math.cos(M) * math.exp(((n - m) + ((M * (n - M)) - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 8.5e-293)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	elseif (n <= 56.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(n - m) + Float64(Float64(M * Float64(n - M)) - l))));
	else
		tmp = Float64(cos(M) * 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 <= 8.5e-293)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	elseif (n <= 56.0)
		tmp = cos(M) * exp(((n - m) + ((M * (n - M)) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 8.5e-293], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 56.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 8.50000000000000044e-293

   1. Initial program 72.9%

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

      1. associate-/l* 72.9%

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

      2. +-commutative 72.9%

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

      3. fabs-sub 72.9%

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

      4. +-commutative 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 70.8%

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

      1. +-commutative 70.8%

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

      2. unpow2 70.8%

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

      3. distribute-rgt-out 79.4%

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

      4. *-commutative 79.4%

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

      5. *-commutative 79.4%

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

   10. Simplified 79.4%

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

   11. Taylor expanded in m around inf 57.2%

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

       1. *-commutative 57.2%

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

   13. Simplified 57.2%

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

   if 8.50000000000000044e-293 < n < 56

   1. Initial program 76.1%

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

      1. associate-/l* 79.1%

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

      2. +-commutative 79.1%

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

      3. fabs-sub 79.1%

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

      4. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in K around 0 89.9%

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

      1. cos-neg 89.9%

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

   7. Simplified 89.9%

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

   8. Taylor expanded in n around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. unpow2 88.9%

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

      3. distribute-rgt-out 88.9%

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

      4. *-commutative 88.9%

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

      5. *-commutative 88.9%

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

   10. Simplified 88.9%

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

   11. Taylor expanded in m around 0 65.4%

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

       1. associate--r+ 65.4%

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

       2. associate-*r* 65.4%

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

       3. neg-mul-1 65.4%

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

       4. cancel-sign-sub 65.4%

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

   13. Simplified 65.4%

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

       1. associate-+l- 65.4%

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

       2. add-sqr-sqrt 45.7%

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

       3. fabs-sqr 45.7%

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

       4. add-sqr-sqrt 80.1%

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

   15. Applied egg-rr 80.1%

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

   if 56 < n

   1. Initial program 81.7%

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

      1. associate-/l* 81.7%

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

      2. +-commutative 81.7%

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

      3. fabs-sub 81.7%

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

      4. +-commutative 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in l around 0 98.4%

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

   9. Taylor expanded in n around inf 98.4%

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

       1. *-commutative 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.5 \cdot 10^{-293}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 56:\\ \;\;\;\;\cos M \cdot e^{\left(n - m\right) + \left(M \cdot \left(n - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -4.0)
   (* (cos M) (exp l))
   (if (<= l 6.6e-24) (* (cos M) (exp (- (pow M 2.0)))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -4.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 6.6e-24) {
		tmp = cos(M) * exp(-pow(M, 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 <= (-4.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 6.6d-24) then
        tmp = cos(m_1) * exp(-(m_1 ** 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 <= -4.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 6.6e-24) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -4.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 6.6e-24:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 6.6e-24)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 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 <= -4.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 6.6e-24)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -4.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e-24], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4

   1. Initial program 74.1%

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

      1. associate-/l* 77.6%

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

      2. +-commutative 77.6%

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

      3. fabs-sub 77.6%

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

      4. +-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in K around 0 94.8%

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

      1. cos-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in n around 0 72.7%

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

      1. +-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. distribute-rgt-out 79.6%

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

      4. *-commutative 79.6%

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

      5. *-commutative 79.6%

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

   10. Simplified 79.6%

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

   11. Taylor expanded in l around inf 20.2%

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

       1. neg-mul-1 20.2%

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

   13. Simplified 20.2%

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

       1. pow1 20.2%

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

       2. add-sqr-sqrt 20.2%

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

       3. sqrt-unprod 20.2%

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

       4. sqr-neg 20.2%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 74.6%

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

   15. Applied egg-rr 74.6%

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

       1. unpow1 74.6%

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

   17. Simplified 74.6%

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

   if -4 < l < 6.59999999999999968e-24

   1. Initial program 75.0%

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

      1. associate-/l* 75.0%

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

      2. +-commutative 75.0%

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

      3. fabs-sub 75.0%

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

      4. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in n around 0 68.8%

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

      1. +-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. distribute-rgt-out 76.4%

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

      4. *-commutative 76.4%

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

      5. *-commutative 76.4%

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

   10. Simplified 76.4%

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

   11. Taylor expanded in M around inf 64.6%

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

       1. mul-1-neg 64.6%

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

   13. Simplified 64.6%

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

   if 6.59999999999999968e-24 < l

   1. Initial program 79.6%

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

      1. associate-/l* 79.6%

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

      2. +-commutative 79.6%

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

      3. fabs-sub 79.6%

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

      4. +-commutative 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. unpow2 83.4%

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

      3. distribute-rgt-out 89.1%

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

      4. *-commutative 89.1%

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

      5. *-commutative 89.1%

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

   10. Simplified 89.1%

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

   11. Taylor expanded in l around inf 96.4%

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

       1. neg-mul-1 96.4%

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

   13. Simplified 96.4%

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

   14. Taylor expanded in M around 0 96.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -730:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-30}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -730.0)
   (* (cos M) (exp l))
   (if (<= l 4.3e-30) (* (cos M) (exp (* M (- n M)))) (/ (cos M) (exp l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -730.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 4.3e-30) {
		tmp = cos(M) * exp((M * (n - M)));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-730.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 4.3d-30) then
        tmp = cos(m_1) * exp((m_1 * (n - m_1)))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -730.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 4.3e-30) {
		tmp = Math.cos(M) * Math.exp((M * (n - M)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -730.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 4.3e-30:
		tmp = math.cos(M) * math.exp((M * (n - M)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -730.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 4.3e-30)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(n - M))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -730.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 4.3e-30)
		tmp = cos(M) * exp((M * (n - M)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -730.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.3e-30], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -730

   1. Initial program 75.4%

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

      1. associate-/l* 78.9%

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

      2. +-commutative 78.9%

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

      3. fabs-sub 78.9%

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

      4. +-commutative 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in n around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 79.2%

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

      4. *-commutative 79.2%

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

      5. *-commutative 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in l around inf 20.5%

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

       1. neg-mul-1 20.5%

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

   13. Simplified 20.5%

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

       1. pow1 20.5%

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

       2. add-sqr-sqrt 20.5%

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

       3. sqrt-unprod 20.5%

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

       4. sqr-neg 20.5%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 75.8%

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

   15. Applied egg-rr 75.8%

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

       1. unpow1 75.8%

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

   17. Simplified 75.8%

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

   if -730 < l < 4.29999999999999966e-30

   1. Initial program 73.9%

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

      1. associate-/l* 73.9%

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

      2. +-commutative 73.9%

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

      3. fabs-sub 73.9%

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

      4. +-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in n around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. unpow2 69.7%

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

      3. distribute-rgt-out 76.8%

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

      4. *-commutative 76.8%

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

      5. *-commutative 76.8%

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

   10. Simplified 76.8%

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

   11. Taylor expanded in m around 0 52.5%

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

       1. associate--r+ 52.5%

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

       2. associate-*r* 52.5%

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

       3. neg-mul-1 52.5%

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

       4. cancel-sign-sub 52.5%

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

   13. Simplified 52.5%

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

   14. Taylor expanded in M around inf 56.2%

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

       1. +-commutative 56.2%

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

       2. mul-1-neg 56.2%

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

       3. unpow2 56.2%

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

       4. distribute-rgt-neg-in 56.2%

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

       5. distribute-lft-in 61.9%

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

       6. sub-neg 61.9%

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

   16. Simplified 61.9%

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

   if 4.29999999999999966e-30 < l

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. +-commutative 80.7%

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

      3. fabs-sub 80.7%

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

      4. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 80.8%

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

      1. +-commutative 80.8%

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

      2. unpow2 80.8%

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

      3. distribute-rgt-out 87.9%

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

      4. *-commutative 87.9%

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

      5. *-commutative 87.9%

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

   10. Simplified 87.9%

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

   11. Taylor expanded in l around inf 93.2%

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

       1. neg-mul-1 93.2%

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

   13. Simplified 93.2%

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

       1. exp-neg 93.2%

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

       2. un-div-inv 93.2%

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

   15. Applied egg-rr 93.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -730:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-30}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-30}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -720.0)
   (* (cos M) (exp l))
   (if (<= l 4.3e-30) (* (cos M) (exp (* M n))) (/ (cos M) (exp l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -720.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 4.3e-30) {
		tmp = cos(M) * exp((M * n));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-720.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 4.3d-30) then
        tmp = cos(m_1) * exp((m_1 * n))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -720.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 4.3e-30) {
		tmp = Math.cos(M) * Math.exp((M * n));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -720.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 4.3e-30:
		tmp = math.cos(M) * math.exp((M * n))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -720.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 4.3e-30)
		tmp = Float64(cos(M) * exp(Float64(M * n)));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -720.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 4.3e-30)
		tmp = cos(M) * exp((M * n));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -720.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.3e-30], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -720

   1. Initial program 75.4%

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

      1. associate-/l* 78.9%

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

      2. +-commutative 78.9%

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

      3. fabs-sub 78.9%

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

      4. +-commutative 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in n around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 79.2%

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

      4. *-commutative 79.2%

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

      5. *-commutative 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in l around inf 20.5%

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

       1. neg-mul-1 20.5%

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

   13. Simplified 20.5%

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

       1. pow1 20.5%

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

       2. add-sqr-sqrt 20.5%

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

       3. sqrt-unprod 20.5%

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

       4. sqr-neg 20.5%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 75.8%

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

   15. Applied egg-rr 75.8%

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

       1. unpow1 75.8%

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

   17. Simplified 75.8%

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

   if -720 < l < 4.29999999999999966e-30

   1. Initial program 73.9%

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

      1. associate-/l* 73.9%

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

      2. +-commutative 73.9%

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

      3. fabs-sub 73.9%

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

      4. +-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in n around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. unpow2 69.7%

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

      3. distribute-rgt-out 76.8%

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

      4. *-commutative 76.8%

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

      5. *-commutative 76.8%

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

   10. Simplified 76.8%

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

   11. Taylor expanded in m around 0 52.5%

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

       1. associate--r+ 52.5%

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

       2. associate-*r* 52.5%

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

       3. neg-mul-1 52.5%

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

       4. cancel-sign-sub 52.5%

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

   13. Simplified 52.5%

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

   14. Taylor expanded in n around inf 33.7%

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

       1. *-commutative 33.7%

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

   16. Simplified 33.7%

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

   if 4.29999999999999966e-30 < l

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. +-commutative 80.7%

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

      3. fabs-sub 80.7%

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

      4. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around 0 80.8%

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

      1. +-commutative 80.8%

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

      2. unpow2 80.8%

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

      3. distribute-rgt-out 87.9%

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

      4. *-commutative 87.9%

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

      5. *-commutative 87.9%

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

   10. Simplified 87.9%

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

   11. Taylor expanded in l around inf 93.2%

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

       1. neg-mul-1 93.2%

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

   13. Simplified 93.2%

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

       1. exp-neg 93.2%

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

       2. un-div-inv 93.2%

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

   15. Applied egg-rr 93.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -720:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-30}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.26 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -1.26e-15) (* (cos M) (exp l)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -1.26e-15) {
		tmp = cos(M) * 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 <= (-1.26d-15)) then
        tmp = cos(m_1) * 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 <= -1.26e-15) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -1.26e-15:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -1.26e-15)
		tmp = Float64(cos(M) * 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 <= -1.26e-15)
		tmp = cos(M) * exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -1.26e-15], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.26e-15

   1. Initial program 75.7%

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

      1. associate-/l* 78.9%

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

      2. +-commutative 78.9%

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

      3. fabs-sub 78.9%

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

      4. +-commutative 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in K around 0 93.7%

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

      1. cos-neg 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in n around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. distribute-rgt-out 79.6%

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

      4. *-commutative 79.6%

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

      5. *-commutative 79.6%

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

   10. Simplified 79.6%

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

   11. Taylor expanded in l around inf 18.8%

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

       1. neg-mul-1 18.8%

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

   13. Simplified 18.8%

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

       1. pow1 18.8%

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

       2. add-sqr-sqrt 18.8%

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

       3. sqrt-unprod 18.8%

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

       4. sqr-neg 18.8%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 68.9%

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

   15. Applied egg-rr 68.9%

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

       1. unpow1 68.9%

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

   17. Simplified 68.9%

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

   if -1.26e-15 < l

   1. Initial program 75.8%

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

      1. associate-/l* 75.8%

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

      2. +-commutative 75.8%

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

      3. fabs-sub 75.8%

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

      4. +-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in n around 0 72.6%

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

      1. +-commutative 72.6%

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

      2. unpow2 72.6%

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

      3. distribute-rgt-out 79.9%

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

      4. *-commutative 79.9%

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

      5. *-commutative 79.9%

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

   10. Simplified 79.9%

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

   11. Taylor expanded in l around inf 33.9%

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

       1. neg-mul-1 33.9%

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

   13. Simplified 33.9%

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

   14. Taylor expanded in M around 0 33.9%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.26 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -8.8e-10) (* (cos M) (exp l)) (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -8.8e-10) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-8.8d-10)) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -8.8e-10) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -8.8e-10:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -8.8e-10)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -8.8e-10)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -8.8e-10], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.7999999999999996e-10

   1. Initial program 75.3%

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

      1. associate-/l* 78.5%

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

      2. +-commutative 78.5%

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

      3. fabs-sub 78.5%

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

      4. +-commutative 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in K around 0 93.6%

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

      1. cos-neg 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in n around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. unpow2 72.8%

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

      3. distribute-rgt-out 79.3%

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

      4. *-commutative 79.3%

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

      5. *-commutative 79.3%

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

   10. Simplified 79.3%

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

   11. Taylor expanded in l around inf 19.1%

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

       1. neg-mul-1 19.1%

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

   13. Simplified 19.1%

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

       1. pow1 19.1%

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

       2. add-sqr-sqrt 19.1%

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

       3. sqrt-unprod 19.1%

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

       4. sqr-neg 19.1%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 69.9%

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

   15. Applied egg-rr 69.9%

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

       1. unpow1 69.9%

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

   17. Simplified 69.9%

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

   if -8.7999999999999996e-10 < l

   1. Initial program 75.9%

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

      1. associate-/l* 75.9%

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

      2. +-commutative 75.9%

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

      3. fabs-sub 75.9%

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

      4. +-commutative 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in n around 0 72.7%

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

      1. +-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. distribute-rgt-out 80.0%

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

      4. *-commutative 80.0%

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

      5. *-commutative 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in l around inf 33.8%

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

       1. neg-mul-1 33.8%

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

   13. Simplified 33.8%

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

       1. exp-neg 33.8%

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

       2. un-div-inv 33.8%

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

   15. Applied egg-rr 33.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ e^{-\ell} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-/l* 76.5%

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

   2. +-commutative 76.5%

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

   3. fabs-sub 76.5%

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

   4. +-commutative 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in K around 0 97.0%

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

   1. cos-neg 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in n around 0 72.7%

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

   1. +-commutative 72.7%

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

   2. unpow2 72.7%

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

   3. distribute-rgt-out 79.8%

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

   4. *-commutative 79.8%

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

   5. *-commutative 79.8%

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

10. Simplified 79.8%

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

11. Taylor expanded in l around inf 30.2%

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

    1. neg-mul-1 30.2%

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

13. Simplified 30.2%

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

14. Taylor expanded in M around 0 29.4%

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

15. Final simplification 29.4%

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

Alternative 14: 7.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-/l* 76.5%

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

   2. +-commutative 76.5%

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

   3. fabs-sub 76.5%

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

   4. +-commutative 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in K around 0 97.0%

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

   1. cos-neg 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in n around 0 72.7%

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

   1. +-commutative 72.7%

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

   2. unpow2 72.7%

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

   3. distribute-rgt-out 79.8%

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

   4. *-commutative 79.8%

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

   5. *-commutative 79.8%

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

10. Simplified 79.8%

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

11. Taylor expanded in l around inf 30.2%

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

    1. neg-mul-1 30.2%

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

13. Simplified 30.2%

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

14. Taylor expanded in l around 0 6.6%

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

15. Final simplification 6.6%

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

Reproduce

herbie shell --seed 2024039 
(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)))))))