Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.7%

Time: 14.7s

Alternatives: 10

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: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% 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 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. *-commutative 76.9%

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

   2. associate-*r/ 76.9%

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

   3. associate--r- 76.9%

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

   4. +-commutative 76.9%

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

   5. associate-+r- 76.9%

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

   6. unsub-neg 76.9%

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

   7. associate--r+ 76.9%

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

   8. +-commutative 76.9%

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

   9. associate--r+ 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 96.3%

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

   1. cos-neg 96.3%

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

6. Simplified 96.3%

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

7. Final simplification 96.3%

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

Alternative 2: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(((m - (n + l)) - ((((m + n) * 0.5d0) - m_1) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 76.9%

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

   2. associate-*r/ 76.9%

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

   3. associate--r- 76.9%

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

   4. +-commutative 76.9%

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

   5. associate-+r- 76.9%

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

   6. unsub-neg 76.9%

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

   7. associate--r+ 76.9%

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

   8. +-commutative 76.9%

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

   9. associate--r+ 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 96.3%

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

   1. cos-neg 96.3%

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

6. Simplified 96.3%

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

   1. expm1-log1p-u 96.3%

      \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

   2. expm1-udef 96.0%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

   3. add-sqr-sqrt 52.1%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

   4. fabs-sqr 52.1%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

   5. add-sqr-sqrt 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

   6. div-inv 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

   7. fma-neg 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

   8. metadata-eval 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

8. Applied egg-rr 96.0%

   \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

   2. expm1-log1p 96.3%

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

   3. associate--l- 96.3%

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

   4. fma-neg 96.3%

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

   5. *-commutative 96.3%

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

   6. +-commutative 96.3%

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 3: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 7.9 \cdot 10^{-171}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;n \leq 35:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 7.9e-171)
   (* (cos M) (exp m))
   (if (<= n 35.0)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* n (* n -0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 7.9e-171) {
		tmp = cos(M) * exp(m);
	} else if (n <= 35.0) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 7.9d-171) then
        tmp = cos(m_1) * exp(m)
    else if (n <= 35.0d0) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 7.9e-171) {
		tmp = Math.cos(M) * Math.exp(m);
	} else if (n <= 35.0) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 7.9e-171:
		tmp = math.cos(M) * math.exp(m)
	elif n <= 35.0:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 7.9e-171)
		tmp = Float64(cos(M) * exp(m));
	elseif (n <= 35.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(n * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 7.9e-171)
		tmp = cos(M) * exp(m);
	elseif (n <= 35.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 7.9e-171], N[(N[Cos[M], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 35.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 7.8999999999999998e-171

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r/ 78.2%

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

      3. associate--r- 78.2%

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

      4. +-commutative 78.2%

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

      5. associate-+r- 78.2%

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

      6. unsub-neg 78.2%

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

      7. associate--r+ 78.2%

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

      8. +-commutative 78.2%

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

      9. associate--r+ 78.2%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in K around 0 95.6%

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

      1. cos-neg 95.6%

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

   6. Simplified 95.6%

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

      1. expm1-log1p-u 95.6%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 95.6%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 65.5%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 65.5%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 95.6%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 95.6%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 95.6%

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

      3. associate--l- 95.6%

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

      4. fma-neg 95.6%

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

      5. *-commutative 95.6%

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

      6. +-commutative 95.6%

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

   10. Simplified 95.6%

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

   11. Taylor expanded in M around inf 64.1%

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

       1. unpow2 64.1%

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

   13. Simplified 64.1%

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

   14. Taylor expanded in m around inf 30.7%

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

   if 7.8999999999999998e-171 < n < 35

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r/ 79.8%

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

      3. associate--r- 79.8%

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

      4. +-commutative 79.8%

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

      5. associate-+r- 79.8%

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

      6. unsub-neg 79.8%

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

      7. associate--r+ 79.8%

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

      8. +-commutative 79.8%

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

      9. associate--r+ 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in K around 0 92.3%

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

      1. cos-neg 92.3%

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

   6. Simplified 92.3%

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

   7. Taylor expanded in M around inf 60.8%

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

      1. unpow2 60.8%

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

      2. neg-mul-1 60.8%

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

      3. distribute-rgt-neg-in 60.8%

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

   9. Simplified 60.8%

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

   if 35 < n

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*r/ 72.3%

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

      3. associate--r- 72.3%

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

      4. +-commutative 72.3%

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

      5. associate-+r- 72.3%

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

      6. unsub-neg 72.3%

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

      7. associate--r+ 72.3%

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

      8. +-commutative 72.3%

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

      9. associate--r+ 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. associate-*l* 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7.9 \cdot 10^{-171}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;n \leq 35:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 4: 74.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp(((m - (n + l)) - (M * 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) * exp(((m - (n + l)) - (m_1 * m_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 76.9%

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

   2. associate-*r/ 76.9%

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

   3. associate--r- 76.9%

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

   4. +-commutative 76.9%

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

   5. associate-+r- 76.9%

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

   6. unsub-neg 76.9%

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

   7. associate--r+ 76.9%

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

   8. +-commutative 76.9%

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

   9. associate--r+ 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 96.3%

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

   1. cos-neg 96.3%

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

6. Simplified 96.3%

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

   1. expm1-log1p-u 96.3%

      \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

   2. expm1-udef 96.0%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

   3. add-sqr-sqrt 52.1%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

   4. fabs-sqr 52.1%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

   5. add-sqr-sqrt 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

   6. div-inv 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

   7. fma-neg 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

   8. metadata-eval 96.0%

      \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

8. Applied egg-rr 96.0%

   \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

   2. expm1-log1p 96.3%

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

   3. associate--l- 96.3%

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

   4. fma-neg 96.3%

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

   5. *-commutative 96.3%

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

   6. +-commutative 96.3%

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

10. Simplified 96.3%

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

11. Taylor expanded in M around inf 70.6%

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

    1. unpow2 70.6%

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

13. Simplified 70.6%

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

14. Final simplification 70.6%

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

Alternative 5: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 9.6 \cdot 10^{-175}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;n \leq 35:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 9.6e-175)
   (* (cos M) (exp m))
   (if (<= n 35.0) (* (cos M) (exp (* M (- M)))) (* (cos M) (exp (- n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 9.6e-175) {
		tmp = cos(M) * exp(m);
	} else if (n <= 35.0) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp(-n);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 9.6d-175) then
        tmp = cos(m_1) * exp(m)
    else if (n <= 35.0d0) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(-n)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 9.6e-175) {
		tmp = Math.cos(M) * Math.exp(m);
	} else if (n <= 35.0) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp(-n);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 9.6e-175:
		tmp = math.cos(M) * math.exp(m)
	elif n <= 35.0:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp(-n)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 9.6e-175)
		tmp = Float64(cos(M) * exp(m));
	elseif (n <= 35.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 9.6e-175)
		tmp = cos(M) * exp(m);
	elseif (n <= 35.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp(-n);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 9.6e-175], N[(N[Cos[M], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 35.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-n)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 9.6e-175

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r/ 78.1%

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

      3. associate--r- 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-+r- 78.1%

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

      6. unsub-neg 78.1%

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

      7. associate--r+ 78.1%

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

      8. +-commutative 78.1%

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

      9. associate--r+ 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in K around 0 95.6%

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

      1. cos-neg 95.6%

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

   6. Simplified 95.6%

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

      1. expm1-log1p-u 95.6%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 95.6%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 65.3%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 65.3%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 95.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 95.6%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 95.6%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 95.6%

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

      3. associate--l- 95.6%

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

      4. fma-neg 95.6%

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

      5. *-commutative 95.6%

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

      6. +-commutative 95.6%

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

   10. Simplified 95.6%

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

   11. Taylor expanded in M around inf 63.9%

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

       1. unpow2 63.9%

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

   13. Simplified 63.9%

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

   14. Taylor expanded in m around inf 30.9%

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

   if 9.6e-175 < n < 35

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*r/ 80.4%

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

      3. associate--r- 80.4%

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

      4. +-commutative 80.4%

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

      5. associate-+r- 80.4%

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

      6. unsub-neg 80.4%

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

      7. associate--r+ 80.4%

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

      8. +-commutative 80.4%

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

      9. associate--r+ 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in K around 0 92.5%

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

      1. cos-neg 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in M around inf 61.9%

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

      1. unpow2 61.9%

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

      2. neg-mul-1 61.9%

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

      3. distribute-rgt-neg-in 61.9%

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

   9. Simplified 61.9%

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

   if 35 < n

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*r/ 72.3%

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

      3. associate--r- 72.3%

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

      4. +-commutative 72.3%

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

      5. associate-+r- 72.3%

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

      6. unsub-neg 72.3%

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

      7. associate--r+ 72.3%

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

      8. +-commutative 72.3%

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

      9. associate--r+ 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 21.5%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 21.5%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate--l- 100.0%

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

      4. fma-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in M around inf 83.3%

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

       1. unpow2 83.3%

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

   13. Simplified 83.3%

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

   14. Taylor expanded in n around inf 97.0%

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

       1. mul-1-neg 97.0%

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

   16. Simplified 97.0%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9.6 \cdot 10^{-175}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;n \leq 35:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]

Alternative 6: 51.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -7e-9) (* (cos M) (exp m)) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7e-9) {
		tmp = cos(M) * exp(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 (m <= (-7d-9)) then
        tmp = cos(m_1) * exp(m)
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7e-9) {
		tmp = Math.cos(M) * Math.exp(m);
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -7e-9:
		tmp = math.cos(M) * math.exp(m)
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -7e-9)
		tmp = Float64(cos(M) * exp(m));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -7e-9)
		tmp = cos(M) * exp(m);
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -7e-9], N[(N[Cos[M], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.9999999999999998e-9

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-*r/ 70.9%

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

      3. associate--r- 70.9%

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

      4. +-commutative 70.9%

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

      5. associate-+r- 70.9%

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

      6. unsub-neg 70.9%

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

      7. associate--r+ 70.9%

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

      8. +-commutative 70.9%

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

      9. associate--r+ 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 12.7%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 12.7%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate--l- 100.0%

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

      4. fma-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in M around inf 91.1%

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

       1. unpow2 91.1%

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

   13. Simplified 91.1%

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

   14. Taylor expanded in m around inf 98.2%

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

   if -6.9999999999999998e-9 < 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. *-commutative 78.6%

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

      2. associate-*r/ 78.6%

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

      3. associate--r- 78.6%

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

      4. +-commutative 78.6%

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

      5. associate-+r- 78.6%

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

      6. unsub-neg 78.6%

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

      7. associate--r+ 78.6%

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

      8. +-commutative 78.6%

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

      9. associate--r+ 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in l around inf 32.9%

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

      1. mul-1-neg 32.9%

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

   9. Simplified 32.9%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 7: 48.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 35:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 35.0) (* (cos M) (exp m)) (* (cos M) (exp (- n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 35.0) {
		tmp = cos(M) * exp(m);
	} else {
		tmp = cos(M) * exp(-n);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 35.0d0) then
        tmp = cos(m_1) * exp(m)
    else
        tmp = cos(m_1) * exp(-n)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 35.0) {
		tmp = Math.cos(M) * Math.exp(m);
	} else {
		tmp = Math.cos(M) * Math.exp(-n);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 35.0:
		tmp = math.cos(M) * math.exp(m)
	else:
		tmp = math.cos(M) * math.exp(-n)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 35.0)
		tmp = Float64(cos(M) * exp(m));
	else
		tmp = Float64(cos(M) * exp(Float64(-n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 35.0)
		tmp = cos(M) * exp(m);
	else
		tmp = cos(M) * exp(-n);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 35.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-n)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 35

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-*r/ 78.5%

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

      3. associate--r- 78.5%

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

      4. +-commutative 78.5%

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

      5. associate-+r- 78.5%

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

      6. unsub-neg 78.5%

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

      7. associate--r+ 78.5%

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

      8. +-commutative 78.5%

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

      9. associate--r+ 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   6. Simplified 95.0%

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

      1. expm1-log1p-u 95.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 94.6%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 62.4%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 62.4%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 94.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 94.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 94.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 94.6%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 94.6%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 95.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 95.0%

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

      3. associate--l- 95.0%

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

      4. fma-neg 95.0%

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

      5. *-commutative 95.0%

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

      6. +-commutative 95.0%

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

   10. Simplified 95.0%

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

   11. Taylor expanded in M around inf 66.2%

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

       1. unpow2 66.2%

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

   13. Simplified 66.2%

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

   14. Taylor expanded in m around inf 29.1%

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

   if 35 < n

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*r/ 72.3%

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

      3. associate--r- 72.3%

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

      4. +-commutative 72.3%

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

      5. associate-+r- 72.3%

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

      6. unsub-neg 72.3%

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

      7. associate--r+ 72.3%

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

      8. +-commutative 72.3%

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

      9. associate--r+ 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 21.5%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 21.5%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate--l- 100.0%

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

      4. fma-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in M around inf 83.3%

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

       1. unpow2 83.3%

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

   13. Simplified 83.3%

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

   14. Taylor expanded in n around inf 97.0%

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

       1. mul-1-neg 97.0%

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

   16. Simplified 97.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 35:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]

Alternative 8: 50.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.85e-37) (* (cos M) (exp m)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.85e-37) {
		tmp = cos(M) * exp(m);
	} 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 (m <= (-1.85d-37)) then
        tmp = cos(m_1) * exp(m)
    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 (m <= -1.85e-37) {
		tmp = Math.cos(M) * Math.exp(m);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.85e-37:
		tmp = math.cos(M) * math.exp(m)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.85e-37)
		tmp = Float64(cos(M) * exp(m));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.85e-37)
		tmp = cos(M) * exp(m);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.85e-37], N[(N[Cos[M], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.85e-37

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*r/ 71.7%

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

      3. associate--r- 71.7%

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

      4. +-commutative 71.7%

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

      5. associate-+r- 71.7%

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

      6. unsub-neg 71.7%

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

      7. associate--r+ 71.7%

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

      8. +-commutative 71.7%

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

      9. associate--r+ 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right)} \]

      3. add-sqr-sqrt 20.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      4. fabs-sqr 20.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\color{blue}{\left(m - n\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\right)} - 1\right) \]

      6. div-inv 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}}\right)} - 1\right) \]

      7. fma-neg 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\color{blue}{\left(\mathsf{fma}\left(m + n, \frac{1}{2}, -M\right)\right)}}^{2}}\right)} - 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \cos M \cdot \left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, \color{blue}{0.5}, -M\right)\right)}^{2}}\right)} - 1\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \cos M \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \cos M \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(m - n\right) - \ell\right) - {\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate--l- 100.0%

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

      4. fma-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in M around inf 85.2%

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

       1. unpow2 85.2%

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

   13. Simplified 85.2%

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

   14. Taylor expanded in m around inf 90.3%

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

   if -1.85e-37 < 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. *-commutative 78.6%

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

      2. associate-*r/ 78.6%

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

      3. associate--r- 78.6%

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

      4. +-commutative 78.6%

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

      5. associate-+r- 78.6%

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

      6. unsub-neg 78.6%

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

      7. associate--r+ 78.6%

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

      8. +-commutative 78.6%

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

      9. associate--r+ 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in K around 0 95.2%

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

      1. cos-neg 95.2%

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

   6. Simplified 95.2%

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

   7. Taylor expanded in l around inf 33.2%

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

      1. mul-1-neg 33.2%

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

   9. Simplified 33.2%

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

   10. Taylor expanded in M around 0 33.2%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 9: 34.1% 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 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. *-commutative 76.9%

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

   2. associate-*r/ 76.9%

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

   3. associate--r- 76.9%

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

   4. +-commutative 76.9%

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

   5. associate-+r- 76.9%

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

   6. unsub-neg 76.9%

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

   7. associate--r+ 76.9%

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

   8. +-commutative 76.9%

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

   9. associate--r+ 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 96.3%

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

   1. cos-neg 96.3%

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

6. Simplified 96.3%

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

7. Taylor expanded in l around inf 32.5%

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

   1. mul-1-neg 32.5%

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

9. Simplified 32.5%

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

10. Taylor expanded in M around 0 32.5%

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

11. Final simplification 32.5%

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

Alternative 10: 7.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative 76.9%

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

   2. associate-*r/ 76.9%

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

   3. associate--r- 76.9%

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

   4. +-commutative 76.9%

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

   5. associate-+r- 76.9%

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

   6. unsub-neg 76.9%

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

   7. associate--r+ 76.9%

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

   8. +-commutative 76.9%

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

   9. associate--r+ 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 96.3%

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

   1. cos-neg 96.3%

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

6. Simplified 96.3%

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

7. Taylor expanded in l around inf 32.5%

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

   1. mul-1-neg 32.5%

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

9. Simplified 32.5%

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

10. Taylor expanded in l around 0 6.1%

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

11. Final simplification 6.1%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023230 
(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)))))))