Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 96.7%

Time: 18.9s

Alternatives: 10

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% 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 81.4%

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

2. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

4. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 96.7% accurate, 1.3× speedup

Math

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

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(Float64(m - 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[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

4. Simplified 98.0%

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

   1. sub-neg 98.0%

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

   2. distribute-neg-out 98.0%

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

   3. div-inv 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-sqr-sqrt 47.9%

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

   6. fabs-sqr 47.9%

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

   7. add-sqr-sqrt 98.0%

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Alternative 3: 88.2% accurate, 1.4× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

4. Simplified 98.0%

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

5. Taylor expanded in m around 0 82.2%

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

   1. associate--r+ 82.2%

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

   2. fabs-sub 82.2%

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

   3. sub-neg 82.2%

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

   4. mul-1-neg 82.2%

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

   5. fabs-neg 82.2%

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

   6. fabs-neg 82.2%

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

   7. mul-1-neg 82.2%

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

   8. sub-neg 82.2%

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

   9. fabs-sub 82.2%

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

   10. unpow1 82.2%

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

   11. sqr-pow 40.2%

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

   12. fabs-sqr 40.2%

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

   13. sqr-pow 89.2%

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

   14. unpow1 89.2%

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

   15. *-commutative 89.2%

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

7. Simplified 89.2%

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

8. Final simplification 89.2%

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

Alternative 4: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-n}\\ \mathbf{if}\;m \leq -54:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -1.25 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -2.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{-290} \lor \neg \left(m \leq 7.1 \cdot 10^{-227}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- n)))))
   (if (<= m -54.0)
     (pow (exp m) (* m -0.25))
     (if (<= m -1.25e-134)
       t_0
       (if (<= m -2.9e-178)
         (/ (cos M) (exp l))
         (if (or (<= m 2.6e-290) (not (<= m 7.1e-227))) t_0 (exp (- l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-n);
	double tmp;
	if (m <= -54.0) {
		tmp = pow(exp(m), (m * -0.25));
	} else if (m <= -1.25e-134) {
		tmp = t_0;
	} else if (m <= -2.9e-178) {
		tmp = cos(M) / exp(l);
	} else if ((m <= 2.6e-290) || !(m <= 7.1e-227)) {
		tmp = t_0;
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(-n)
    if (m <= (-54.0d0)) then
        tmp = exp(m) ** (m * (-0.25d0))
    else if (m <= (-1.25d-134)) then
        tmp = t_0
    else if (m <= (-2.9d-178)) then
        tmp = cos(m_1) / exp(l)
    else if ((m <= 2.6d-290) .or. (.not. (m <= 7.1d-227))) then
        tmp = t_0
    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 t_0 = Math.cos(M) * Math.exp(-n);
	double tmp;
	if (m <= -54.0) {
		tmp = Math.pow(Math.exp(m), (m * -0.25));
	} else if (m <= -1.25e-134) {
		tmp = t_0;
	} else if (m <= -2.9e-178) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if ((m <= 2.6e-290) || !(m <= 7.1e-227)) {
		tmp = t_0;
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-n)
	tmp = 0
	if m <= -54.0:
		tmp = math.pow(math.exp(m), (m * -0.25))
	elif m <= -1.25e-134:
		tmp = t_0
	elif m <= -2.9e-178:
		tmp = math.cos(M) / math.exp(l)
	elif (m <= 2.6e-290) or not (m <= 7.1e-227):
		tmp = t_0
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-n)))
	tmp = 0.0
	if (m <= -54.0)
		tmp = exp(m) ^ Float64(m * -0.25);
	elseif (m <= -1.25e-134)
		tmp = t_0;
	elseif (m <= -2.9e-178)
		tmp = Float64(cos(M) / exp(l));
	elseif ((m <= 2.6e-290) || !(m <= 7.1e-227))
		tmp = t_0;
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-n);
	tmp = 0.0;
	if (m <= -54.0)
		tmp = exp(m) ^ (m * -0.25);
	elseif (m <= -1.25e-134)
		tmp = t_0;
	elseif (m <= -2.9e-178)
		tmp = cos(M) / exp(l);
	elseif ((m <= 2.6e-290) || ~((m <= 7.1e-227)))
		tmp = t_0;
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-n)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -54.0], N[Power[N[Exp[m], $MachinePrecision], N[(m * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1.25e-134], t$95$0, If[LessEqual[m, -2.9e-178], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[m, 2.6e-290], N[Not[LessEqual[m, 7.1e-227]], $MachinePrecision]], t$95$0, N[Exp[(-l)], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -54

   1. Initial program 69.8%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in m around inf 98.1%

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

      1. *-commutative 68.0%

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

      2. unpow2 68.0%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in M around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. associate-*r* 98.1%

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

      4. exp-prod 98.1%

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

   10. Simplified 98.1%

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

   if -54 < m < -1.2500000000000001e-134 or -2.8999999999999998e-178 < m < 2.60000000000000001e-290 or 7.0999999999999996e-227 < m

   1. Initial program 83.7%

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

   2. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   4. Simplified 97.1%

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

      1. sub-neg 97.1%

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

      2. distribute-neg-out 97.1%

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

      3. div-inv 97.1%

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

      4. metadata-eval 97.1%

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

      5. add-sqr-sqrt 60.6%

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

      6. fabs-sqr 60.6%

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

      7. add-sqr-sqrt 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in M around inf 72.5%

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

      1. unpow2 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in n around inf 33.4%

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

   if -1.2500000000000001e-134 < m < -2.8999999999999998e-178

   1. Initial program 83.3%

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

   2. Taylor expanded in l around inf 59.1%

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

      1. mul-1-neg 59.1%

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

   4. Simplified 59.1%

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

   5. Taylor expanded in K around 0 67.6%

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

      1. cos-neg 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in l around -inf 67.6%

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

      1. neg-mul-1 67.6%

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

      2. exp-neg 67.6%

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

      3. associate-*r/ 67.6%

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

      4. *-rgt-identity 67.6%

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

   10. Simplified 67.6%

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

   if 2.60000000000000001e-290 < m < 7.0999999999999996e-227

   1. Initial program 93.3%

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

   2. Taylor expanded in l around inf 34.8%

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

      1. mul-1-neg 34.8%

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

   4. Simplified 34.8%

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

   5. Taylor expanded in K around 0 35.0%

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

      1. cos-neg 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in M around 0 35.0%

      \[\leadsto \color{blue}{e^{-\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq -1.25 \cdot 10^{-134}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \mathbf{elif}\;m \leq -2.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{-290} \lor \neg \left(m \leq 7.1 \cdot 10^{-227}\right):\\ \;\;\;\;\cos M \cdot e^{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 5: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{m}\\ \mathbf{if}\;n \leq -6.6 \cdot 10^{-286}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-295}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-264}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 720:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp m))))
   (if (<= n -6.6e-286)
     t_0
     (if (<= n 1.65e-295)
       (exp (- l))
       (if (<= n 1.1e-264)
         t_0
         (if (<= n 720.0)
           (* (cos M) (exp (* M (- M))))
           (* (cos M) (exp (- n)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(m);
	double tmp;
	if (n <= -6.6e-286) {
		tmp = t_0;
	} else if (n <= 1.65e-295) {
		tmp = exp(-l);
	} else if (n <= 1.1e-264) {
		tmp = t_0;
	} else if (n <= 720.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(m)
    if (n <= (-6.6d-286)) then
        tmp = t_0
    else if (n <= 1.65d-295) then
        tmp = exp(-l)
    else if (n <= 1.1d-264) then
        tmp = t_0
    else if (n <= 720.0d0) 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 t_0 = Math.cos(M) * Math.exp(m);
	double tmp;
	if (n <= -6.6e-286) {
		tmp = t_0;
	} else if (n <= 1.65e-295) {
		tmp = Math.exp(-l);
	} else if (n <= 1.1e-264) {
		tmp = t_0;
	} else if (n <= 720.0) {
		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):
	t_0 = math.cos(M) * math.exp(m)
	tmp = 0
	if n <= -6.6e-286:
		tmp = t_0
	elif n <= 1.65e-295:
		tmp = math.exp(-l)
	elif n <= 1.1e-264:
		tmp = t_0
	elif n <= 720.0:
		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)
	t_0 = Float64(cos(M) * exp(m))
	tmp = 0.0
	if (n <= -6.6e-286)
		tmp = t_0;
	elseif (n <= 1.65e-295)
		tmp = exp(Float64(-l));
	elseif (n <= 1.1e-264)
		tmp = t_0;
	elseif (n <= 720.0)
		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)
	t_0 = cos(M) * exp(m);
	tmp = 0.0;
	if (n <= -6.6e-286)
		tmp = t_0;
	elseif (n <= 1.65e-295)
		tmp = exp(-l);
	elseif (n <= 1.1e-264)
		tmp = t_0;
	elseif (n <= 720.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp(-n);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.6e-286], t$95$0, If[LessEqual[n, 1.65e-295], N[Exp[(-l)], $MachinePrecision], If[LessEqual[n, 1.1e-264], t$95$0, If[LessEqual[n, 720.0], 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 4 regimes

2. if n < -6.5999999999999997e-286 or 1.6499999999999999e-295 < n < 1.09999999999999997e-264

   1. Initial program 83.5%

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

   2. Taylor expanded in K around 0 98.2%

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

      1. cos-neg 98.2%

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

   4. Simplified 98.2%

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

      1. sub-neg 98.2%

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

      2. distribute-neg-out 98.2%

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

      3. div-inv 98.2%

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

      4. metadata-eval 98.2%

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

      5. add-sqr-sqrt 75.1%

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

      6. fabs-sqr 75.1%

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

      7. add-sqr-sqrt 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in M around inf 64.6%

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

      1. unpow2 64.6%

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

   9. Simplified 64.6%

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

   10. Taylor expanded in m around inf 29.1%

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

       1. mul-1-neg 29.1%

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

   12. Simplified 29.1%

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

   if -6.5999999999999997e-286 < n < 1.6499999999999999e-295

   1. Initial program 100.0%

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

   2. Taylor expanded in l around inf 83.9%

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

      1. mul-1-neg 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in K around 0 83.9%

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

      1. cos-neg 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in M around 0 83.9%

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

   if 1.09999999999999997e-264 < n < 720

   1. Initial program 91.9%

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

   2. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   4. Simplified 95.3%

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

   5. Taylor expanded in M around inf 67.3%

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

      1. mul-1-neg 67.3%

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

      2. unpow2 67.3%

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

      3. distribute-rgt-neg-in 67.3%

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

   7. Simplified 67.3%

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

   if 720 < n

   1. Initial program 66.2%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-out 100.0%

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

      3. div-inv 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-sqr-sqrt 9.2%

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

      6. fabs-sqr 9.2%

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

      7. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in M around inf 92.4%

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

      1. unpow2 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in n around inf 100.0%

       \[\leadsto \cos M \cdot e^{-\color{blue}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{-286}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-295}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-264}:\\ \;\;\;\;\cos M \cdot e^{m}\\ \mathbf{elif}\;n \leq 720:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]

Alternative 6: 75.2% accurate, 2.0× speedup

Math

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

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(Float64(m - 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[(N[(m - n), $MachinePrecision] - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

4. Simplified 98.0%

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

   1. sub-neg 98.0%

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

   2. distribute-neg-out 98.0%

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

   3. div-inv 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-sqr-sqrt 47.9%

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

   6. fabs-sqr 47.9%

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

   7. add-sqr-sqrt 98.0%

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

6. Applied egg-rr 98.0%

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

7. Taylor expanded in M around inf 77.6%

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

   1. unpow2 77.6%

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

9. Simplified 77.6%

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

10. Final simplification 77.6%

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

Alternative 7: 52.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 3.5e-6) (exp (- l)) (* (cos M) (exp (- n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3.5e-6) {
		tmp = exp(-l);
	} 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 <= 3.5d-6) then
        tmp = exp(-l)
    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 <= 3.5e-6) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp(-n);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 3.5e-6:
		tmp = math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp(-n)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 3.5e-6)
		tmp = exp(Float64(-l));
	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 <= 3.5e-6)
		tmp = exp(-l);
	else
		tmp = cos(M) * exp(-n);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 3.5e-6], N[Exp[(-l)], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-n)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.49999999999999995e-6

   1. Initial program 86.5%

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

   2. Taylor expanded in l around inf 39.1%

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

      1. mul-1-neg 39.1%

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

   4. Simplified 39.1%

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

   5. Taylor expanded in K around 0 42.2%

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

      1. cos-neg 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in M around 0 42.8%

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

   if 3.49999999999999995e-6 < n

   1. Initial program 67.2%

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

   2. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   4. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-out 100.0%

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

      3. div-inv 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-sqr-sqrt 9.0%

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

      6. fabs-sqr 9.0%

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

      7. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in M around inf 92.7%

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

      1. unpow2 92.7%

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

   9. Simplified 92.7%

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

   10. Taylor expanded in n around inf 97.1%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-n}\\ \end{array} \]

Alternative 8: 36.2% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in l around inf 34.7%

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

   1. mul-1-neg 34.7%

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

4. Simplified 34.7%

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

5. Taylor expanded in K around 0 38.7%

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

   1. cos-neg 38.7%

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

7. Simplified 38.7%

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

8. Taylor expanded in M around 0 39.1%

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

9. Final simplification 39.1%

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

Alternative 9: 7.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in m around inf 41.6%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative 41.6%

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

   2. unpow2 41.6%

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

4. Simplified 41.6%

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

5. Taylor expanded in m around 0 9.6%

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

   1. *-commutative 9.6%

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

   2. associate-*l* 9.6%

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

7. Simplified 9.6%

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

8. Taylor expanded in K around 0 9.8%

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

   1. cos-neg 9.8%

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

10. Simplified 9.8%

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

11. Final simplification 9.8%

    \[\leadsto \cos M \]

Alternative 10: 6.9% accurate, 141.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in l around inf 34.7%

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

   1. mul-1-neg 34.7%

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

4. Simplified 34.7%

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

5. Taylor expanded in m around inf 37.4%

   \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(K \cdot m\right)\right)} \cdot e^{-\ell} \]
6. Step-by-step derivation

   1. *-commutative 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

7. Simplified 37.4%

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

8. Taylor expanded in l around 0 9.4%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot m\right)\right) + -1 \cdot \left(\ell \cdot \cos \left(0.5 \cdot \left(K \cdot m\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 9.4%

      \[\leadsto \cos \color{blue}{\left(\left(0.5 \cdot K\right) \cdot m\right)} + -1 \cdot \left(\ell \cdot \cos \left(0.5 \cdot \left(K \cdot m\right)\right)\right) \]

   2. *-commutative 9.4%

      \[\leadsto \cos \color{blue}{\left(m \cdot \left(0.5 \cdot K\right)\right)} + -1 \cdot \left(\ell \cdot \cos \left(0.5 \cdot \left(K \cdot m\right)\right)\right) \]

   3. *-lft-identity 9.4%

      \[\leadsto \color{blue}{1 \cdot \cos \left(m \cdot \left(0.5 \cdot K\right)\right)} + -1 \cdot \left(\ell \cdot \cos \left(0.5 \cdot \left(K \cdot m\right)\right)\right) \]

   4. associate-*r* 9.4%

      \[\leadsto 1 \cdot \cos \left(m \cdot \left(0.5 \cdot K\right)\right) + \color{blue}{\left(-1 \cdot \ell\right) \cdot \cos \left(0.5 \cdot \left(K \cdot m\right)\right)} \]

   5. neg-mul-1 9.4%

      \[\leadsto 1 \cdot \cos \left(m \cdot \left(0.5 \cdot K\right)\right) + \color{blue}{\left(-\ell\right)} \cdot \cos \left(0.5 \cdot \left(K \cdot m\right)\right) \]

   6. associate-*r* 9.4%

      \[\leadsto 1 \cdot \cos \left(m \cdot \left(0.5 \cdot K\right)\right) + \left(-\ell\right) \cdot \cos \color{blue}{\left(\left(0.5 \cdot K\right) \cdot m\right)} \]

   7. *-commutative 9.4%

      \[\leadsto 1 \cdot \cos \left(m \cdot \left(0.5 \cdot K\right)\right) + \left(-\ell\right) \cdot \cos \color{blue}{\left(m \cdot \left(0.5 \cdot K\right)\right)} \]

   8. distribute-rgt-out 9.4%

      \[\leadsto \color{blue}{\cos \left(m \cdot \left(0.5 \cdot K\right)\right) \cdot \left(1 + \left(-\ell\right)\right)} \]

   9. *-commutative 9.4%

      \[\leadsto \cos \color{blue}{\left(\left(0.5 \cdot K\right) \cdot m\right)} \cdot \left(1 + \left(-\ell\right)\right) \]

   10. associate-*r* 9.4%

       \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(K \cdot m\right)\right)} \cdot \left(1 + \left(-\ell\right)\right) \]

   11. *-commutative 9.4%

       \[\leadsto \cos \left(0.5 \cdot \color{blue}{\left(m \cdot K\right)}\right) \cdot \left(1 + \left(-\ell\right)\right) \]

   12. sub-neg 9.4%

       \[\leadsto \cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot \color{blue}{\left(1 - \ell\right)} \]

10. Simplified 9.4%

    \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(m \cdot K\right)\right) \cdot \left(1 - \ell\right)} \]

11. Taylor expanded in m around 0 9.7%

    \[\leadsto \color{blue}{1 - \ell} \]

12. Final simplification 9.7%

    \[\leadsto 1 - \ell \]

Reproduce

herbie shell --seed 2023278 
(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)))))))