Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.5%

Time: 19.4s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := t_0 - \ell\\ t_2 := {\left(\frac{m + n}{2} - M\right)}^{2}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{t_1 - t_2} \leq \infty:\\ \;\;\;\;\cos \left(\frac{K}{{\left(\sqrt[3]{\frac{2}{m + n}}\right)}^{3}} - M\right) \cdot e^{t_0 - \left(\ell + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_1 - \frac{m + n}{\frac{4}{m + n}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m)))
        (t_1 (- t_0 l))
        (t_2 (pow (- (/ (+ m n) 2.0) M) 2.0)))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- t_1 t_2))) INFINITY)
     (*
      (cos (- (/ K (pow (cbrt (/ 2.0 (+ m n))) 3.0)) M))
      (exp (- t_0 (+ l t_2))))
     (exp (- t_1 (/ (+ m n) (/ 4.0 (+ m n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = t_0 - l;
	double t_2 = pow((((m + n) / 2.0) - M), 2.0);
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * exp((t_1 - t_2))) <= ((double) INFINITY)) {
		tmp = cos(((K / pow(cbrt((2.0 / (m + n))), 3.0)) - M)) * exp((t_0 - (l + t_2)));
	} else {
		tmp = exp((t_1 - ((m + n) / (4.0 / (m + n)))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double t_1 = t_0 - l;
	double t_2 = Math.pow((((m + n) / 2.0) - M), 2.0);
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((t_1 - t_2))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.cos(((K / Math.pow(Math.cbrt((2.0 / (m + n))), 3.0)) - M)) * Math.exp((t_0 - (l + t_2)));
	} else {
		tmp = Math.exp((t_1 - ((m + n) / (4.0 / (m + n)))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(t_0 - l)
	t_2 = Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(t_1 - t_2))) <= Inf)
		tmp = Float64(cos(Float64(Float64(K / (cbrt(Float64(2.0 / Float64(m + n))) ^ 3.0)) - M)) * exp(Float64(t_0 - Float64(l + t_2))));
	else
		tmp = exp(Float64(t_1 - Float64(Float64(m + n) / Float64(4.0 / Float64(m + n)))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - l), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$1 - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Cos[N[(N[(K / N[Power[N[Power[N[(2.0 / N[(m + n), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$1 - N[(N[(m + n), $MachinePrecision] / N[(4.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 96.0%

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

   3. Simplified 96.0%

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

      1. add-cube-cbrt 97.3%

         \[\leadsto \cos \left(\frac{K}{\color{blue}{\left(\sqrt[3]{\frac{2}{m + n}} \cdot \sqrt[3]{\frac{2}{m + n}}\right) \cdot \sqrt[3]{\frac{2}{m + n}}}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

      2. pow3 97.3%

         \[\leadsto \cos \left(\frac{K}{\color{blue}{{\left(\sqrt[3]{\frac{2}{m + n}}\right)}^{3}}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

   5. Applied egg-rr 97.3%

      \[\leadsto \cos \left(\frac{K}{\color{blue}{{\left(\sqrt[3]{\frac{2}{m + n}}\right)}^{3}}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) 2) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) 2) M) 2)) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   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 M around 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. fabs-sub 100.0%

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

      3. +-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. metadata-eval 100.0%

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

       4. swap-sqr 100.0%

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

       5. metadata-eval 100.0%

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

       6. div-inv 100.0%

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

       7. metadata-eval 100.0%

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

       8. div-inv 100.0%

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

       9. clear-num 100.0%

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

       10. metadata-eval 100.0%

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

       11. frac-times 100.0%

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

       12. *-commutative 100.0%

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

       13. metadata-eval 100.0%

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

       14. *-un-lft-identity 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. associate-*r/ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{\color{blue}{4}}{n + m}}} \]

       4. +-commutative 100.0%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{4}{\color{blue}{m + n}}}} \]

   13. Simplified 100.0%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\frac{m + n}{\frac{4}{m + n}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;\cos \left(\frac{K}{{\left(\sqrt[3]{\frac{2}{m + n}}\right)}^{3}} - M\right) \cdot e^{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m + n}}}\\ \end{array} \]

Alternative 2: 96.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Simplified 73.9%

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

4. Taylor expanded in K around 0 96.1%

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

   1. cos-neg 96.1%

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

6. Simplified 96.1%

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

7. Taylor expanded in M around 0 96.3%

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

8. Final simplification 96.3%

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

Alternative 3: 87.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;e^{\left(t_0 - \ell\right) - \frac{m + n}{\frac{4}{m + n}}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -0.43)
     (exp (- (- t_0 l) (/ (+ m n) (/ 4.0 (+ m n)))))
     (exp (+ t_0 (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -0.43) {
		tmp = exp(((t_0 - l) - ((m + n) / (4.0 / (m + n)))));
	} else {
		tmp = exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - 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 = abs((n - m))
    if (m <= (-0.43d0)) then
        tmp = exp(((t_0 - l) - ((m + n) / (4.0d0 / (m + n)))))
    else
        tmp = exp((t_0 + ((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - 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.abs((n - m));
	double tmp;
	if (m <= -0.43) {
		tmp = Math.exp(((t_0 - l) - ((m + n) / (4.0 / (m + n)))));
	} else {
		tmp = Math.exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -0.43:
		tmp = math.exp(((t_0 - l) - ((m + n) / (4.0 / (m + n)))))
	else:
		tmp = math.exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -0.43)
		tmp = exp(Float64(Float64(t_0 - l) - Float64(Float64(m + n) / Float64(4.0 / Float64(m + n)))));
	else
		tmp = exp(Float64(t_0 + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -0.43)
		tmp = exp(((t_0 - l) - ((m + n) / (4.0 / (m + n)))));
	else
		tmp = exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	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, -0.43], N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[(N[(m + n), $MachinePrecision] / N[(4.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 + N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -0.429999999999999993

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   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 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. fabs-sub 100.0%

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

      3. +-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. metadata-eval 100.0%

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

       4. swap-sqr 100.0%

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

       5. metadata-eval 100.0%

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

       6. div-inv 100.0%

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

       7. metadata-eval 100.0%

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

       8. div-inv 100.0%

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

       9. clear-num 100.0%

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

       10. metadata-eval 100.0%

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

       11. frac-times 100.0%

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

       12. *-commutative 100.0%

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

       13. metadata-eval 100.0%

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

       14. *-un-lft-identity 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. associate-*r/ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{\color{blue}{4}}{n + m}}} \]

       4. +-commutative 100.0%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{4}{\color{blue}{m + n}}}} \]

   13. Simplified 100.0%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\frac{m + n}{\frac{4}{m + n}}}} \]

   if -0.429999999999999993 < m

   1. Initial program 75.4%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in K around 0 94.8%

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

      1. cos-neg 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in M around 0 95.0%

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

   8. Taylor expanded in m around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. distribute-rgt-out 83.7%

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

      4. *-commutative 83.7%

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

      5. *-commutative 83.7%

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

   10. Simplified 83.7%

       \[\leadsto 1 \cdot e^{\left(\left(-\color{blue}{\left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right)}\right) - \ell\right) + \left|n - m\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m + n}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \]

Alternative 4: 70.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ \mathbf{if}\;m \leq -2.5 \cdot 10^{+40}:\\ \;\;\;\;e^{t_0 - \frac{m + n}{\frac{4}{m}}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \frac{m + n}{\frac{4}{n}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l)))
   (if (<= m -2.5e+40)
     (exp (- t_0 (/ (+ m n) (/ 4.0 m))))
     (exp (- t_0 (/ (+ m n) (/ 4.0 n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double tmp;
	if (m <= -2.5e+40) {
		tmp = exp((t_0 - ((m + n) / (4.0 / m))));
	} else {
		tmp = exp((t_0 - ((m + n) / (4.0 / 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 = abs((n - m)) - l
    if (m <= (-2.5d+40)) then
        tmp = exp((t_0 - ((m + n) / (4.0d0 / m))))
    else
        tmp = exp((t_0 - ((m + n) / (4.0d0 / 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.abs((n - m)) - l;
	double tmp;
	if (m <= -2.5e+40) {
		tmp = Math.exp((t_0 - ((m + n) / (4.0 / m))));
	} else {
		tmp = Math.exp((t_0 - ((m + n) / (4.0 / n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	tmp = 0
	if m <= -2.5e+40:
		tmp = math.exp((t_0 - ((m + n) / (4.0 / m))))
	else:
		tmp = math.exp((t_0 - ((m + n) / (4.0 / n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	tmp = 0.0
	if (m <= -2.5e+40)
		tmp = exp(Float64(t_0 - Float64(Float64(m + n) / Float64(4.0 / m))));
	else
		tmp = exp(Float64(t_0 - Float64(Float64(m + n) / Float64(4.0 / n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	tmp = 0.0;
	if (m <= -2.5e+40)
		tmp = exp((t_0 - ((m + n) / (4.0 / m))));
	else
		tmp = exp((t_0 - ((m + n) / (4.0 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -2.5e+40], N[Exp[N[(t$95$0 - N[(N[(m + n), $MachinePrecision] / N[(4.0 / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - N[(N[(m + n), $MachinePrecision] / N[(4.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.50000000000000002e40

   1. Initial program 66.7%

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

   3. Simplified 66.7%

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

   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 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. fabs-sub 100.0%

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

      3. +-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. metadata-eval 100.0%

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

       4. swap-sqr 100.0%

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

       5. metadata-eval 100.0%

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

       6. div-inv 100.0%

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

       7. metadata-eval 100.0%

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

       8. div-inv 100.0%

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

       9. clear-num 100.0%

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

       10. metadata-eval 100.0%

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

       11. frac-times 100.0%

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

       12. *-commutative 100.0%

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

       13. metadata-eval 100.0%

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

       14. *-un-lft-identity 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. associate-*r/ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{\color{blue}{4}}{n + m}}} \]

       4. +-commutative 100.0%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{4}{\color{blue}{m + n}}}} \]

   13. Simplified 100.0%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\frac{m + n}{\frac{4}{m + n}}}} \]

   14. Taylor expanded in m around inf 93.1%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\color{blue}{\frac{4}{m}}}} \]

   if -2.50000000000000002e40 < m

   1. Initial program 75.9%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in M around 0 82.5%

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

      1. associate--r+ 82.5%

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

      2. fabs-sub 82.5%

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

      3. +-commutative 82.5%

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

   9. Simplified 82.5%

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

       1. *-commutative 82.5%

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

       2. unpow2 82.5%

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

       3. metadata-eval 82.5%

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

       4. swap-sqr 82.5%

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

       5. metadata-eval 82.5%

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

       6. div-inv 82.5%

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

       7. metadata-eval 82.5%

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

       8. div-inv 82.5%

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

       9. clear-num 82.5%

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

       10. metadata-eval 82.5%

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

       11. frac-times 82.5%

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

       12. *-commutative 82.5%

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

       13. metadata-eval 82.5%

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

       14. *-un-lft-identity 82.5%

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

   11. Applied egg-rr 82.5%

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

       1. +-commutative 82.5%

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

       2. associate-*r/ 82.5%

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

       3. metadata-eval 82.5%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{\color{blue}{4}}{n + m}}} \]

       4. +-commutative 82.5%

          \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{4}{\color{blue}{m + n}}}} \]

   13. Simplified 82.5%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\frac{m + n}{\frac{4}{m + n}}}} \]

   14. Taylor expanded in m around 0 66.2%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\color{blue}{\frac{4}{n}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{+40}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{n}}}\\ \end{array} \]

Alternative 5: 86.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m + n}}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (- (fabs (- n m)) l) (/ (+ m n) (/ 4.0 (+ m n))))))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((n - m)) - l) - ((m + n) / (4.0 / (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 = exp(((abs((n - m)) - l) - ((m + n) / (4.0d0 / (m + n)))))
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) - ((m + n) / (4.0 / (m + n)))));
}

Python

def code(K, m, n, M, l):
	return math.exp(((math.fabs((n - m)) - l) - ((m + n) / (4.0 / (m + n)))))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m + n) / Float64(4.0 / Float64(m + n)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((n - m)) - l) - ((m + n) / (4.0 / (m + n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m + n), $MachinePrecision] / N[(4.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.8%

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

3. Simplified 73.9%

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

4. Taylor expanded in K around 0 96.1%

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

   1. cos-neg 96.1%

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

6. Simplified 96.1%

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

7. Taylor expanded in M around 0 86.4%

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

   1. associate--r+ 86.4%

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

   2. fabs-sub 86.4%

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

   3. +-commutative 86.4%

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

9. Simplified 86.4%

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

    1. *-commutative 86.4%

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

    2. unpow2 86.4%

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

    3. metadata-eval 86.4%

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

    4. swap-sqr 86.4%

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

    5. metadata-eval 86.4%

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

    6. div-inv 86.4%

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

    7. metadata-eval 86.4%

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

    8. div-inv 86.4%

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

    9. clear-num 86.4%

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

    10. metadata-eval 86.4%

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

    11. frac-times 86.4%

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

    12. *-commutative 86.4%

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

    13. metadata-eval 86.4%

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

    14. *-un-lft-identity 86.4%

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

11. Applied egg-rr 86.4%

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

    1. +-commutative 86.4%

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

    2. associate-*r/ 86.4%

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

    3. metadata-eval 86.4%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{\color{blue}{4}}{n + m}}} \]

    4. +-commutative 86.4%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{4}{\color{blue}{m + n}}}} \]

13. Simplified 86.4%

    \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\frac{m + n}{\frac{4}{m + n}}}} \]

14. Final simplification 86.4%

    \[\leadsto e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m + n}}} \]

Alternative 6: 60.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m}}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (- (fabs (- n m)) l) (/ (+ m n) (/ 4.0 m)))))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((n - m)) - l) - ((m + n) / (4.0 / 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 = exp(((abs((n - m)) - l) - ((m + n) / (4.0d0 / m))))
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) - ((m + n) / (4.0 / m))));
}

Python

def code(K, m, n, M, l):
	return math.exp(((math.fabs((n - m)) - l) - ((m + n) / (4.0 / m))))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m + n) / Float64(4.0 / m))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((n - m)) - l) - ((m + n) / (4.0 / m))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m + n), $MachinePrecision] / N[(4.0 / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.8%

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

3. Simplified 73.9%

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

4. Taylor expanded in K around 0 96.1%

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

   1. cos-neg 96.1%

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

6. Simplified 96.1%

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

7. Taylor expanded in M around 0 86.4%

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

   1. associate--r+ 86.4%

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

   2. fabs-sub 86.4%

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

   3. +-commutative 86.4%

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

9. Simplified 86.4%

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

    1. *-commutative 86.4%

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

    2. unpow2 86.4%

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

    3. metadata-eval 86.4%

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

    4. swap-sqr 86.4%

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

    5. metadata-eval 86.4%

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

    6. div-inv 86.4%

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

    7. metadata-eval 86.4%

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

    8. div-inv 86.4%

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

    9. clear-num 86.4%

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

    10. metadata-eval 86.4%

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

    11. frac-times 86.4%

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

    12. *-commutative 86.4%

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

    13. metadata-eval 86.4%

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

    14. *-un-lft-identity 86.4%

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

11. Applied egg-rr 86.4%

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

    1. +-commutative 86.4%

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

    2. associate-*r/ 86.4%

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

    3. metadata-eval 86.4%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{\color{blue}{4}}{n + m}}} \]

    4. +-commutative 86.4%

       \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\frac{4}{\color{blue}{m + n}}}} \]

13. Simplified 86.4%

    \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\frac{m + n}{\frac{4}{m + n}}}} \]

14. Taylor expanded in m around inf 63.4%

    \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - \frac{m + n}{\color{blue}{\frac{4}{m}}}} \]

15. Final simplification 63.4%

    \[\leadsto e^{\left(\left|n - m\right| - \ell\right) - \frac{m + n}{\frac{4}{m}}} \]

Reproduce

herbie shell --seed 2023301 
(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)))))))