Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.9%

Time: 11.0s

Alternatives: 10

Speedup: 1.6×

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: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := e^{\left(t\_0 - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \cdot \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_0 - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m)))
        (t_1
         (*
          (exp (- (- t_0 l) (pow (- (/ (+ n m) 2.0) M) 2.0)))
          (cos (- (/ (* (+ n m) K) 2.0) M)))))
   (if (<= t_1 INFINITY)
     t_1
     (* (cos M) (exp (- t_0 (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = exp(((t_0 - l) - pow((((n + m) / 2.0) - M), 2.0))) * cos(((((n + m) * K) / 2.0) - M));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = cos(M) * exp((t_0 - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(exp(Float64(Float64(t_0 - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))) * cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))));
	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[(N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 97.4%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

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

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;e^{\left(t\_0 - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \cdot \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \leq -0.5:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(\mathsf{fma}\left(-K, {\left(\frac{-2}{n + m}\right)}^{-1}, -M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_0 - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<=
        (*
         (exp (- (- t_0 l) (pow (- (/ (+ n m) 2.0) M) 2.0)))
         (cos (- (/ (* (+ n m) K) 2.0) M)))
        -0.5)
     (* (exp (- l)) (cos (fma (- K) (pow (/ -2.0 (+ n m)) -1.0) (- M))))
     (* (cos M) (exp (- t_0 (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((exp(((t_0 - l) - pow((((n + m) / 2.0) - M), 2.0))) * cos(((((n + m) * K) / 2.0) - M))) <= -0.5) {
		tmp = exp(-l) * cos(fma(-K, pow((-2.0 / (n + m)), -1.0), -M));
	} else {
		tmp = cos(M) * exp((t_0 - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (Float64(exp(Float64(Float64(t_0 - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))) * cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M))) <= -0.5)
		tmp = Float64(exp(Float64(-l)) * cos(fma(Float64(-K), (Float64(-2.0 / Float64(n + m)) ^ -1.0), Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[Exp[(-l)], $MachinePrecision] * N[Cos[N[((-K) * N[Power[N[(-2.0 / N[(n + m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < -0.5

   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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 71.2

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

   7. Applied rewrites 71.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-+.f64 N/A

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

      16. lift-neg.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -0.5 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 77.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \cdot \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right) \leq -0.5:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(\mathsf{fma}\left(-K, {\left(\frac{-2}{n + m}\right)}^{-1}, -M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{\left(n + m\right) \cdot K}{2} - M\right)\\ t_1 := \left|n - m\right|\\ \mathbf{if}\;e^{\left(t\_1 - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \cdot t\_0 \leq -0.5:\\ \;\;\;\;e^{-\ell} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t\_1 - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (/ (* (+ n m) K) 2.0) M))) (t_1 (fabs (- n m))))
   (if (<= (* (exp (- (- t_1 l) (pow (- (/ (+ n m) 2.0) M) 2.0))) t_0) -0.5)
     (* (exp (- l)) t_0)
     (* (cos M) (exp (- t_1 (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((((n + m) * K) / 2.0) - M));
	double t_1 = fabs((n - m));
	double tmp;
	if ((exp(((t_1 - l) - pow((((n + m) / 2.0) - M), 2.0))) * t_0) <= -0.5) {
		tmp = exp(-l) * t_0;
	} else {
		tmp = cos(M) * exp((t_1 - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(Float64(Float64(n + m) * K) / 2.0) - M))
	t_1 = abs(Float64(n - m))
	tmp = 0.0
	if (Float64(exp(Float64(Float64(t_1 - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))) * t_0) <= -0.5)
		tmp = Float64(exp(Float64(-l)) * t_0);
	else
		tmp = Float64(cos(M) * exp(Float64(t_1 - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(N[(t$95$1 - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], -0.5], N[(N[Exp[(-l)], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < -0.5

   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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -0.5 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 77.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

4. Final simplification 97.5%

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

Alternative 4: 87.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, m, -M\right)\\ t_1 := \left|n - m\right|\\ \mathbf{if}\;n \leq 9.8 \cdot 10^{-26}:\\ \;\;\;\;e^{t\_1 - \mathsf{fma}\left(t\_0, t\_0, \ell\right)} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1 - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fma 0.5 m (- M))) (t_1 (fabs (- n m))))
   (if (<= n 9.8e-26)
     (* (exp (- t_1 (fma t_0 t_0 l))) (cos M))
     (exp (- t_1 (fma 0.25 (pow (+ n m) 2.0) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fma(0.5, m, -M);
	double t_1 = fabs((n - m));
	double tmp;
	if (n <= 9.8e-26) {
		tmp = exp((t_1 - fma(t_0, t_0, l))) * cos(M);
	} else {
		tmp = exp((t_1 - fma(0.25, pow((n + m), 2.0), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = fma(0.5, m, Float64(-M))
	t_1 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 9.8e-26)
		tmp = Float64(exp(Float64(t_1 - fma(t_0, t_0, l))) * cos(M));
	else
		tmp = exp(Float64(t_1 - fma(0.25, (Float64(n + m) ^ 2.0), l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * m + (-M)), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 9.8e-26], N[(N[Exp[N[(t$95$1 - N[(t$95$0 * t$95$0 + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$1 - N[(0.25 * N[Power[N[(n + m), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if n < 9.7999999999999998e-26

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in n around 0

      \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, m, -M\right), \frac{1}{2} \cdot m - M, \ell\right)} \cdot \cos M \]
   10. Step-by-step derivation

   11. Applied rewrites 84.2%

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

   if 9.7999999999999998e-26 < n

   1. Initial program 70.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.7%

      \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9.8 \cdot 10^{-26}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(\mathsf{fma}\left(0.5, m, -M\right), \mathsf{fma}\left(0.5, m, -M\right), \ell\right)} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -3.1 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -3.1e+99)
     t_0
     (if (<= M 1.8e+18)
       (exp (- (fabs (- n m)) (fma 0.25 (pow (+ n m) 2.0) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -3.1e+99) {
		tmp = t_0;
	} else if (M <= 1.8e+18) {
		tmp = exp((fabs((n - m)) - fma(0.25, pow((n + m), 2.0), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -3.1e+99)
		tmp = t_0;
	elseif (M <= 1.8e+18)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(0.25, (Float64(n + m) ^ 2.0), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -3.1e+99], t$95$0, If[LessEqual[M, 1.8e+18], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[Power[N[(n + m), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -3.1000000000000001e99 or 1.8e18 < M

   1. Initial program 79.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 98.2%

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

   if -3.1000000000000001e99 < M < 1.8e18

   1. Initial program 75.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.1%

      \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{+99}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, {\left(n + m\right)}^{2}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;e^{\left(-0.25 \cdot m\right) \cdot m}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;1 \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -1.1e-286)
   (exp (* (* -0.25 m) m))
   (if (<= n 53.0)
     (* 1.0 (exp (- (fabs (- n m)) (* M M))))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.1e-286) {
		tmp = exp(((-0.25 * m) * m));
	} else if (n <= 53.0) {
		tmp = 1.0 * exp((fabs((n - m)) - (M * M)));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-1.1d-286)) then
        tmp = exp((((-0.25d0) * m) * m))
    else if (n <= 53.0d0) then
        tmp = 1.0d0 * exp((abs((n - m)) - (m_1 * m_1)))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -1.1e-286) {
		tmp = Math.exp(((-0.25 * m) * m));
	} else if (n <= 53.0) {
		tmp = 1.0 * Math.exp((Math.abs((n - m)) - (M * M)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -1.1e-286:
		tmp = math.exp(((-0.25 * m) * m))
	elif n <= 53.0:
		tmp = 1.0 * math.exp((math.fabs((n - m)) - (M * M)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -1.1e-286)
		tmp = exp(Float64(Float64(-0.25 * m) * m));
	elseif (n <= 53.0)
		tmp = Float64(1.0 * exp(Float64(abs(Float64(n - m)) - Float64(M * M))));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -1.1e-286)
		tmp = exp(((-0.25 * m) * m));
	elseif (n <= 53.0)
		tmp = 1.0 * exp((abs((n - m)) - (M * M)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -1.1e-286], N[Exp[N[(N[(-0.25 * m), $MachinePrecision] * m), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 53.0], N[(1.0 * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1e-286

   1. Initial program 76.5%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.2%

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

   if -1.1e-286 < n < 53

   1. Initial program 90.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in M around inf

      \[\leadsto e^{\left|m - n\right| - {M}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - M \cdot M} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 62.2%

       \[\leadsto e^{\left|m - n\right| - M \cdot M} \cdot 1 \]

   if 53 < n

   1. Initial program 69.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;e^{\left(-0.25 \cdot m\right) \cdot m}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;1 \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 135:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 135.0)
   (exp (- (fabs (- n m)) (fma 0.25 (* m m) l)))
   (exp (* -0.25 (* n n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 135.0) {
		tmp = exp((fabs((n - m)) - fma(0.25, (m * m), l)));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 135.0)
		tmp = exp(Float64(abs(Float64(n - m)) - fma(0.25, Float64(m * m), l)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 135.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 135

   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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.1%

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

   9. Taylor expanded in n around 0

      \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, {m}^{2}, \ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.8%

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

   if 135 < n

   1. Initial program 69.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 135:\\ \;\;\;\;e^{\left|n - m\right| - \mathsf{fma}\left(0.25, m \cdot m, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.75 \cdot 10^{-82}:\\ \;\;\;\;e^{\left(-0.25 \cdot m\right) \cdot m}\\ \mathbf{elif}\;n \leq 135:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2.75e-82)
   (exp (* (* -0.25 m) m))
   (if (<= n 135.0) (exp (- l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.75e-82) {
		tmp = exp(((-0.25 * m) * m));
	} else if (n <= 135.0) {
		tmp = exp(-l);
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 2.75d-82) then
        tmp = exp((((-0.25d0) * m) * m))
    else if (n <= 135.0d0) then
        tmp = exp(-l)
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.75e-82) {
		tmp = Math.exp(((-0.25 * m) * m));
	} else if (n <= 135.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.75e-82:
		tmp = math.exp(((-0.25 * m) * m))
	elif n <= 135.0:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2.75e-82)
		tmp = exp(Float64(Float64(-0.25 * m) * m));
	elseif (n <= 135.0)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2.75e-82)
		tmp = exp(((-0.25 * m) * m));
	elseif (n <= 135.0)
		tmp = exp(-l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2.75e-82], N[Exp[N[(N[(-0.25 * m), $MachinePrecision] * m), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 135.0], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.7499999999999999e-82

   1. Initial program 79.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.1%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

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

   if 2.7499999999999999e-82 < n < 135

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.3%

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

   9. Taylor expanded in l around inf

      \[\leadsto e^{-1 \cdot \ell} \]
   10. Step-by-step derivation

   11. Applied rewrites 51.0%

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

   if 135 < n

   1. Initial program 69.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.75 \cdot 10^{-82}:\\ \;\;\;\;e^{\left(-0.25 \cdot m\right) \cdot m}\\ \mathbf{elif}\;n \leq 135:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-0.25 \cdot m\right) \cdot m}\\ \mathbf{if}\;m \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* -0.25 m) m))))
   (if (<= m -5.5e-8) t_0 (if (<= m 54.0) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((-0.25 * m) * m));
	double tmp;
	if (m <= -5.5e-8) {
		tmp = t_0;
	} else if (m <= 54.0) {
		tmp = exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((((-0.25d0) * m) * m))
    if (m <= (-5.5d-8)) then
        tmp = t_0
    else if (m <= 54.0d0) then
        tmp = exp(-l)
    else
        tmp = t_0
    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.exp(((-0.25 * m) * m));
	double tmp;
	if (m <= -5.5e-8) {
		tmp = t_0;
	} else if (m <= 54.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((-0.25 * m) * m))
	tmp = 0
	if m <= -5.5e-8:
		tmp = t_0
	elif m <= 54.0:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(-0.25 * m) * m))
	tmp = 0.0
	if (m <= -5.5e-8)
		tmp = t_0;
	elseif (m <= 54.0)
		tmp = exp(Float64(-l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((-0.25 * m) * m));
	tmp = 0.0;
	if (m <= -5.5e-8)
		tmp = t_0;
	elseif (m <= 54.0)
		tmp = exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(-0.25 * m), $MachinePrecision] * m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -5.5e-8], t$95$0, If[LessEqual[m, 54.0], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.5000000000000003e-8 or 54 < m

   1. Initial program 68.1%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.8%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.1%

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

   if -5.5000000000000003e-8 < m < 54

   1. Initial program 85.5%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in M around 0

      \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.6%

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

   9. Taylor expanded in l around inf

      \[\leadsto e^{-1 \cdot \ell} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.7%

       \[\leadsto e^{-\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 35.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in M around 0

   \[\leadsto e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 87.0%

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

9. Taylor expanded in l around inf

   \[\leadsto e^{-1 \cdot \ell} \]
10. Step-by-step derivation

11. Applied rewrites 38.6%

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

Reproduce

herbie shell --seed 2024248 
(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)))))))