Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 96.5%

Time: 24.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((n - m)) - (pow((((m + n) / 2.0) - M), 2.0) + 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 = cos(m_1) * exp((abs((n - m)) - (((((m + n) / 2.0d0) - m_1) ** 2.0d0) + l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. fabs-sub 75.3%

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

   4. associate-/l* 74.9%

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

   5. +-commutative 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

6. Simplified 97.5%

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

7. Final simplification 97.5%

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

Alternative 2: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq -2.55 \cdot 10^{-163}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 38000000000000:\\ \;\;\;\;e^{t_0 - \left(\ell + M \cdot M\right)} \cdot \cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n -2.55e-163)
     (* (cos M) (exp (+ t_0 (* -0.25 (* m m)))))
     (if (<= n 38000000000000.0)
       (* (exp (- t_0 (+ l (* M M)))) (cos (- (* 0.5 (* n K)) M)))
       (exp (fma -0.25 (* n n) t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= -2.55e-163) {
		tmp = cos(M) * exp((t_0 + (-0.25 * (m * m))));
	} else if (n <= 38000000000000.0) {
		tmp = exp((t_0 - (l + (M * M)))) * cos(((0.5 * (n * K)) - M));
	} else {
		tmp = exp(fma(-0.25, (n * n), t_0));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= -2.55e-163)
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(-0.25 * Float64(m * m)))));
	elseif (n <= 38000000000000.0)
		tmp = Float64(exp(Float64(t_0 - Float64(l + Float64(M * M)))) * cos(Float64(Float64(0.5 * Float64(n * K)) - M)));
	else
		tmp = exp(fma(-0.25, Float64(n * n), t_0));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -2.55e-163], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 38000000000000.0], N[(N[Exp[N[(t$95$0 - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.54999999999999995e-163

   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%

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

      2. +-commutative 72.3%

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

      3. fabs-sub 72.3%

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

      4. associate-/l* 71.2%

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

      5. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in K around 0 98.9%

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

      1. cos-neg 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in m around inf 49.7%

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

      1. unpow2 49.7%

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

   9. Simplified 49.7%

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

   if -2.54999999999999995e-163 < n < 3.8e13

   1. Initial program 82.9%

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

      1. +-commutative 82.9%

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

      2. +-commutative 82.9%

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

      3. fabs-sub 82.9%

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

      4. associate-/l* 82.1%

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

      5. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in M around inf 71.9%

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

      1. unpow2 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in m around 0 79.1%

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

      1. unpow2 79.1%

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

   9. Simplified 79.1%

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

   if 3.8e13 < n

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. fabs-sub 68.5%

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

      4. associate-/l* 69.9%

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

      5. +-commutative 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 83.8%

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

      1. *-commutative 83.8%

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

      2. unpow2 83.8%

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

      3. associate-*l* 83.8%

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

   9. Simplified 83.8%

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

   10. Taylor expanded in M around 0 83.8%

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

       1. exp-sum 0.0%

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

       2. sub-neg 0.0%

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

       3. mul-1-neg 0.0%

          \[\leadsto e^{-0.25 \cdot {n}^{2}} \cdot e^{\left|n + \color{blue}{-1 \cdot m}\right|} \]

       4. exp-sum 83.8%

          \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2} + \left|n + -1 \cdot m\right|}} \]

       5. fma-def 83.8%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(-0.25, {n}^{2}, \left|n + -1 \cdot m\right|\right)}} \]

       6. unpow2 83.8%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, \color{blue}{n \cdot n}, \left|n + -1 \cdot m\right|\right)} \]

       7. mul-1-neg 83.8%

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

       8. sub-neg 83.8%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|\color{blue}{n - m}\right|\right)} \]

   12. Simplified 83.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.55 \cdot 10^{-163}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 38000000000000:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + M \cdot M\right)} \cdot \cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}\\ \end{array} \]

Alternative 3: 57.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-140}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -5.4e+17)
     (* (cos M) (exp (+ t_0 (* -0.25 (* m m)))))
     (if (<= m 7.5e-140)
       (* (cos M) (exp (- t_0 (* M M))))
       (* (cos M) (exp (+ t_0 (* n (* n -0.25)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -5.4e+17) {
		tmp = cos(M) * exp((t_0 + (-0.25 * (m * m))));
	} else if (m <= 7.5e-140) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * exp((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-5.4d+17)) then
        tmp = cos(m_1) * exp((t_0 + ((-0.25d0) * (m * m))))
    else if (m <= 7.5d-140) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * exp((t_0 + (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 t_0 = Math.abs((n - m));
	double tmp;
	if (m <= -5.4e+17) {
		tmp = Math.cos(M) * Math.exp((t_0 + (-0.25 * (m * m))));
	} else if (m <= 7.5e-140) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + (n * (n * -0.25))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -5.4e+17:
		tmp = math.cos(M) * math.exp((t_0 + (-0.25 * (m * m))))
	elif m <= 7.5e-140:
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + (n * (n * -0.25))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -5.4e+17)
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(-0.25 * Float64(m * m)))));
	elseif (m <= 7.5e-140)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(n * Float64(n * -0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -5.4e+17)
		tmp = cos(M) * exp((t_0 + (-0.25 * (m * m))));
	elseif (m <= 7.5e-140)
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 + (n * (n * -0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -5.4e+17], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.5e-140], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.4e17

   1. Initial program 71.2%

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

      1. +-commutative 71.2%

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

      2. +-commutative 71.2%

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

      3. fabs-sub 71.2%

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

      4. associate-/l* 71.2%

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

      5. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 86.7%

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

      1. unpow2 86.7%

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

   9. Simplified 86.7%

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

   if -5.4e17 < m < 7.4999999999999998e-140

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. +-commutative 80.8%

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

      3. fabs-sub 80.8%

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

      4. associate-/l* 80.0%

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

      5. +-commutative 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in M around inf 62.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}} + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. mul-1-neg 62.9%

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

      2. unpow2 62.9%

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

      3. distribute-rgt-neg-out 62.9%

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

   9. Simplified 62.9%

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

   if 7.4999999999999998e-140 < m

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. +-commutative 71.8%

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

      3. fabs-sub 71.8%

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

      4. associate-/l* 71.8%

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

      5. +-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in n around inf 41.4%

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

      1. *-commutative 41.4%

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

      2. unpow2 41.4%

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

      3. associate-*l* 41.4%

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

   9. Simplified 41.4%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-140}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 4: 68.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq -2 \cdot 10^{+84} \lor \neg \left(n \leq 200000000000\right):\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= n -2e+84) (not (<= n 200000000000.0)))
     (exp (fma -0.25 (* n n) t_0))
     (* (cos M) (exp (- t_0 (* M M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((n <= -2e+84) || !(n <= 200000000000.0)) {
		tmp = exp(fma(-0.25, (n * n), t_0));
	} else {
		tmp = cos(M) * exp((t_0 - (M * M)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((n <= -2e+84) || !(n <= 200000000000.0))
		tmp = exp(fma(-0.25, Float64(n * n), t_0));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[n, -2e+84], N[Not[LessEqual[n, 200000000000.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.00000000000000012e84 or 2e11 < 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. Step-by-step derivation

      1. +-commutative 67.2%

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

      2. +-commutative 67.2%

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

      3. fabs-sub 67.2%

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

      4. associate-/l* 67.2%

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

      5. +-commutative 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 86.8%

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

      1. *-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. associate-*l* 86.8%

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

   9. Simplified 86.8%

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

   10. Taylor expanded in M around 0 86.8%

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

       1. exp-sum 0.0%

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

       2. sub-neg 0.0%

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

       3. mul-1-neg 0.0%

          \[\leadsto e^{-0.25 \cdot {n}^{2}} \cdot e^{\left|n + \color{blue}{-1 \cdot m}\right|} \]

       4. exp-sum 86.8%

          \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2} + \left|n + -1 \cdot m\right|}} \]

       5. fma-def 86.8%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(-0.25, {n}^{2}, \left|n + -1 \cdot m\right|\right)}} \]

       6. unpow2 86.8%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, \color{blue}{n \cdot n}, \left|n + -1 \cdot m\right|\right)} \]

       7. mul-1-neg 86.8%

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

       8. sub-neg 86.8%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|\color{blue}{n - m}\right|\right)} \]

   12. Simplified 86.8%

       \[\leadsto \color{blue}{e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}} \]

   if -2.00000000000000012e84 < n < 2e11

   1. Initial program 82.2%

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

      1. +-commutative 82.2%

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

      2. +-commutative 82.2%

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

      3. fabs-sub 82.2%

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

      4. associate-/l* 81.6%

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

      5. +-commutative 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in K around 0 95.4%

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

      1. cos-neg 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in M around inf 60.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}} + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. mul-1-neg 60.3%

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

      2. unpow2 60.3%

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

      3. distribute-rgt-neg-out 60.3%

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

   9. Simplified 60.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+84} \lor \neg \left(n \leq 200000000000\right):\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - M \cdot M}\\ \end{array} \]

Alternative 5: 57.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -1.56 \cdot 10^{+18}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{-135}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -1.56e+18)
     (* (cos M) (exp (+ t_0 (* -0.25 (* m m)))))
     (if (<= m 1.65e-135)
       (* (cos M) (exp (- t_0 (* M M))))
       (exp (fma -0.25 (* n n) t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -1.56e+18) {
		tmp = cos(M) * exp((t_0 + (-0.25 * (m * m))));
	} else if (m <= 1.65e-135) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = exp(fma(-0.25, (n * n), t_0));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -1.56e+18)
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(-0.25 * Float64(m * m)))));
	elseif (m <= 1.65e-135)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = exp(fma(-0.25, Float64(n * n), t_0));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -1.56e+18], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65e-135], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.56e18

   1. Initial program 71.2%

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

      1. +-commutative 71.2%

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

      2. +-commutative 71.2%

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

      3. fabs-sub 71.2%

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

      4. associate-/l* 71.2%

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

      5. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 86.7%

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

      1. unpow2 86.7%

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

   9. Simplified 86.7%

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

   if -1.56e18 < m < 1.65e-135

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. +-commutative 80.0%

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

      3. fabs-sub 80.0%

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

      4. associate-/l* 79.2%

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

      5. +-commutative 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in K around 0 94.6%

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

      1. cos-neg 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in M around inf 62.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}} + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. mul-1-neg 62.3%

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

      2. unpow2 62.3%

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

      3. distribute-rgt-neg-out 62.3%

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

   9. Simplified 62.3%

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

   if 1.65e-135 < m

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

      2. +-commutative 72.6%

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

      3. fabs-sub 72.6%

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

      4. associate-/l* 72.6%

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

      5. +-commutative 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in n around inf 41.8%

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

      1. *-commutative 41.8%

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

      2. unpow2 41.8%

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

      3. associate-*l* 41.8%

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

   9. Simplified 41.8%

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

   10. Taylor expanded in M around 0 41.8%

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

       1. exp-sum 1.2%

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

       2. sub-neg 1.2%

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

       3. mul-1-neg 1.2%

          \[\leadsto e^{-0.25 \cdot {n}^{2}} \cdot e^{\left|n + \color{blue}{-1 \cdot m}\right|} \]

       4. exp-sum 41.8%

          \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2} + \left|n + -1 \cdot m\right|}} \]

       5. fma-def 41.8%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(-0.25, {n}^{2}, \left|n + -1 \cdot m\right|\right)}} \]

       6. unpow2 41.8%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, \color{blue}{n \cdot n}, \left|n + -1 \cdot m\right|\right)} \]

       7. mul-1-neg 41.8%

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

       8. sub-neg 41.8%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|\color{blue}{n - m}\right|\right)} \]

   12. Simplified 41.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.56 \cdot 10^{+18}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{-135}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}\\ \end{array} \]

Alternative 6: 60.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{+36} \lor \neg \left(n \leq 0.5\right):\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= n -1.15e+36) (not (<= n 0.5)))
     (exp (fma -0.25 (* n n) t_0))
     (exp (- t_0 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((n <= -1.15e+36) || !(n <= 0.5)) {
		tmp = exp(fma(-0.25, (n * n), t_0));
	} else {
		tmp = exp((t_0 - l));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((n <= -1.15e+36) || !(n <= 0.5))
		tmp = exp(fma(-0.25, Float64(n * n), t_0));
	else
		tmp = exp(Float64(t_0 - l));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[n, -1.15e+36], N[Not[LessEqual[n, 0.5]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - l), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.14999999999999998e36 or 0.5 < n

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. +-commutative 68.8%

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

      3. fabs-sub 68.8%

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

      4. associate-/l* 68.8%

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

      5. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 84.6%

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

      1. *-commutative 84.6%

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

      2. unpow2 84.6%

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

      3. associate-*l* 84.6%

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

   9. Simplified 84.6%

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

   10. Taylor expanded in M around 0 84.6%

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

       1. exp-sum 0.1%

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

       2. sub-neg 0.1%

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

       3. mul-1-neg 0.1%

          \[\leadsto e^{-0.25 \cdot {n}^{2}} \cdot e^{\left|n + \color{blue}{-1 \cdot m}\right|} \]

       4. exp-sum 84.6%

          \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2} + \left|n + -1 \cdot m\right|}} \]

       5. fma-def 84.6%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(-0.25, {n}^{2}, \left|n + -1 \cdot m\right|\right)}} \]

       6. unpow2 84.6%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, \color{blue}{n \cdot n}, \left|n + -1 \cdot m\right|\right)} \]

       7. mul-1-neg 84.6%

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

       8. sub-neg 84.6%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|\color{blue}{n - m}\right|\right)} \]

   12. Simplified 84.6%

       \[\leadsto \color{blue}{e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}} \]

   if -1.14999999999999998e36 < n < 0.5

   1. Initial program 81.8%

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

      1. +-commutative 81.8%

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

      2. +-commutative 81.8%

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

      3. fabs-sub 81.8%

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

      4. associate-/l* 81.1%

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

      5. +-commutative 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in M around inf 64.9%

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

      1. unpow2 64.9%

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

   6. Simplified 64.9%

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

   7. Taylor expanded in M around 0 28.9%

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

   8. Taylor expanded in K around 0 31.5%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+36} \lor \neg \left(n \leq 0.5\right):\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \ell}\\ \end{array} \]

Alternative 7: 59.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq -1.5 \cdot 10^{+66} \lor \neg \left(n \leq 5\right):\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= n -1.5e+66) (not (<= n 5.0)))
     (exp (fma -0.25 (* n n) t_0))
     (* (cos M) (exp (- t_0 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((n <= -1.5e+66) || !(n <= 5.0)) {
		tmp = exp(fma(-0.25, (n * n), t_0));
	} else {
		tmp = cos(M) * exp((t_0 - l));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((n <= -1.5e+66) || !(n <= 5.0))
		tmp = exp(fma(-0.25, Float64(n * n), t_0));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[n, -1.5e+66], N[Not[LessEqual[n, 5.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.50000000000000001e66 or 5 < n

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. +-commutative 68.0%

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

      3. fabs-sub 68.0%

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

      4. associate-/l* 68.0%

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

      5. +-commutative 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 86.3%

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

      1. *-commutative 86.3%

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

      2. unpow2 86.3%

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

      3. associate-*l* 86.3%

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

   9. Simplified 86.3%

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

   10. Taylor expanded in M around 0 86.3%

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

       1. exp-sum 0.0%

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

       2. sub-neg 0.0%

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

       3. mul-1-neg 0.0%

          \[\leadsto e^{-0.25 \cdot {n}^{2}} \cdot e^{\left|n + \color{blue}{-1 \cdot m}\right|} \]

       4. exp-sum 86.3%

          \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2} + \left|n + -1 \cdot m\right|}} \]

       5. fma-def 86.3%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(-0.25, {n}^{2}, \left|n + -1 \cdot m\right|\right)}} \]

       6. unpow2 86.3%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, \color{blue}{n \cdot n}, \left|n + -1 \cdot m\right|\right)} \]

       7. mul-1-neg 86.3%

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

       8. sub-neg 86.3%

          \[\leadsto e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|\color{blue}{n - m}\right|\right)} \]

   12. Simplified 86.3%

       \[\leadsto \color{blue}{e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}} \]

   if -1.50000000000000001e66 < n < 5

   1. Initial program 81.8%

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

      1. +-commutative 81.8%

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

      2. +-commutative 81.8%

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

      3. fabs-sub 81.8%

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

      4. associate-/l* 81.2%

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

      5. +-commutative 81.2%

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

   3. Simplified 81.2%

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

      1. associate-/l* 81.8%

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

      2. expm1-log1p-u 58.6%

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

      3. div-inv 58.6%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(K \cdot \left(m + n\right)\right) \cdot \frac{1}{2}}\right)\right) - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

      4. *-commutative 58.6%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(m + n\right) \cdot K\right)} \cdot \frac{1}{2}\right)\right) - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

      5. associate-*l* 58.6%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(m + n\right) \cdot \left(K \cdot \frac{1}{2}\right)}\right)\right) - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

      6. metadata-eval 58.6%

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

   5. Applied egg-rr 58.6%

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

   6. Taylor expanded in l around inf 23.5%

      \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right)\right) - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
   7. Step-by-step derivation

      1. neg-mul-1 23.5%

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

   8. Simplified 23.5%

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

   9. Taylor expanded in K around 0 30.9%

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

       1. cos-neg 30.9%

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

       2. sub-neg 30.9%

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

       3. mul-1-neg 30.9%

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

       4. mul-1-neg 30.9%

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

       5. sub-neg 30.9%

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

       6. fabs-sub 30.9%

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

   11. Simplified 30.9%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+66} \lor \neg \left(n \leq 5\right):\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, n \cdot n, \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \ell}\\ \end{array} \]

Alternative 8: 23.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \ell} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - 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((abs((n - m)) - l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. fabs-sub 75.3%

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

   4. associate-/l* 74.9%

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

   5. +-commutative 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in M around inf 49.5%

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

   1. unpow2 49.5%

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

6. Simplified 49.5%

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

7. Taylor expanded in M around 0 19.2%

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

8. Taylor expanded in K around 0 23.1%

   \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell}} \]

9. Final simplification 23.1%

   \[\leadsto e^{\left|n - m\right| - \ell} \]

Reproduce

herbie shell --seed 2023275 
(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)))))))