Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.0%

Time: 13.3s

Alternatives: 10

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.7% 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: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := e^{\left(t\_0 - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_2 := K \cdot \left(m + n\right)\\ t_3 := 0.5 \cdot t\_2\\ \mathbf{if}\;\cos \left(\frac{t\_2}{2} - M\right) \cdot t\_1 \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(M, \sin t\_3, \cos t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n)))
        (t_1 (exp (- (- t_0 l) (pow (- (/ (+ m n) 2.0) M) 2.0))))
        (t_2 (* K (+ m n)))
        (t_3 (* 0.5 t_2)))
   (if (<= (* (cos (- (/ t_2 2.0) M)) t_1) INFINITY)
     (* t_1 (fma M (sin t_3) (cos t_3)))
     (exp (- t_0 (fma 0.25 (* (+ m n) (+ m n)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = exp(((t_0 - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double t_2 = K * (m + n);
	double t_3 = 0.5 * t_2;
	double tmp;
	if ((cos(((t_2 / 2.0) - M)) * t_1) <= ((double) INFINITY)) {
		tmp = t_1 * fma(M, sin(t_3), cos(t_3));
	} else {
		tmp = exp((t_0 - fma(0.25, ((m + n) * (m + n)), l)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = exp(Float64(Float64(t_0 - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	t_2 = Float64(K * Float64(m + n))
	t_3 = Float64(0.5 * t_2)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(t_2 / 2.0) - M)) * t_1) <= Inf)
		tmp = Float64(t_1 * fma(M, sin(t_3), cos(t_3)));
	else
		tmp = exp(Float64(t_0 - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(t$95$2 / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(t$95$1 * N[(M * N[Sin[t$95$3], $MachinePrecision] + N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $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 95.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 96.2

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

   5. Simplified 96.2%

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

   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

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

      1. cos-neg N/A

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

      2. lower-cos.f64 98.1

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

   5. Simplified 98.1%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto e^{\color{blue}{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

      3. fabs-sub N/A

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{-1 \cdot m}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      6. lower-fabs.f64 N/A

         \[\leadsto e^{\color{blue}{\left|n + -1 \cdot m\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n - m}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      9. lower--.f64 N/A

         \[\leadsto e^{\left|\color{blue}{n - m}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      10. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} + \ell\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 98.1

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

   8. Simplified 98.1%

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

4. Final simplification 96.6%

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

Alternative 2: 95.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -126:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 45:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -126.0)
     t_0
     (if (<= M 45.0)
       (exp (- (fabs (- m n)) (fma 0.25 (* (+ m n) (+ m n)) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -126.0) {
		tmp = t_0;
	} else if (M <= 45.0) {
		tmp = exp((fabs((m - n)) - fma(0.25, ((m + n) * (m + n)), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -126.0)
		tmp = t_0;
	elseif (M <= 45.0)
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -126.0], t$95$0, If[LessEqual[M, 45.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -126 or 45 < M

   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 M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 96.9

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

   11. Simplified 96.9%

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

   if -126 < M < 45

   1. Initial program 75.1%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 92.6

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

   5. Simplified 92.6%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto e^{\color{blue}{\left|m - n\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)}} \]

      3. fabs-sub N/A

         \[\leadsto e^{\color{blue}{\left|n - m\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n + \left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{-1 \cdot m}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      6. lower-fabs.f64 N/A

         \[\leadsto e^{\color{blue}{\left|n + -1 \cdot m\right|} - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      7. mul-1-neg N/A

         \[\leadsto e^{\left|n + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      8. sub-neg N/A

         \[\leadsto e^{\left|\color{blue}{n - m}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      9. lower--.f64 N/A

         \[\leadsto e^{\left|\color{blue}{n - m}\right| - \left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right)} \]

      10. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} + \ell\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      12. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      14. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      16. +-commutative N/A

          \[\leadsto e^{\left|n - m\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      17. lower-+.f64 92.6

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

   8. Simplified 92.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -126:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 45:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(m + n\right) - M\\ e^{\left|m - n\right| - \mathsf{fma}\left(t\_0, t\_0, \ell\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ m n)) M))) (exp (- (fabs (- m n)) (fma t_0 t_0 l)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (m + n)) - M;
	return exp((fabs((m - n)) - fma(t_0, t_0, l)));
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(m + n)) - M)
	return exp(Float64(abs(Float64(m - n)) - fma(t_0, t_0, l)))
end

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in M around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 76.3

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

5. Simplified 76.3%

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

6. Taylor expanded in K around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. fabs-sub N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower-fabs.f64 N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 95.0

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 4: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -530000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.45 \cdot 10^{-296}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -530000000.0)
   (exp (* (* m m) -0.25))
   (if (<= m -2.45e-296)
     (exp (* M (- M)))
     (exp (- (fabs (- m n)) (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -530000000.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -2.45e-296) {
		tmp = exp((M * -M));
	} else {
		tmp = exp((fabs((m - n)) - (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 (m <= (-530000000.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-2.45d-296)) then
        tmp = exp((m_1 * -m_1))
    else
        tmp = exp((abs((m - n)) - (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 (m <= -530000000.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -2.45e-296) {
		tmp = Math.exp((M * -M));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -530000000.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -2.45e-296:
		tmp = math.exp((M * -M))
	else:
		tmp = math.exp((math.fabs((m - n)) - (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -530000000.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -2.45e-296)
		tmp = exp(Float64(M * Float64(-M)));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -530000000.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -2.45e-296)
		tmp = exp((M * -M));
	else
		tmp = exp((abs((m - n)) - (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -530000000.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -2.45e-296], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.3e8

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 65.0

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

   5. Simplified 65.0%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 98.4

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

   11. Simplified 98.4%

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

   if -5.3e8 < m < -2.4499999999999999e-296

   1. Initial program 81.6%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 81.6

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

   5. Simplified 81.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 90.6

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

   8. Simplified 90.6%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 62.6

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

   11. Simplified 62.6%

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

   if -2.4499999999999999e-296 < m

   1. Initial program 78.9%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 78.9

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

   5. Simplified 78.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 94.8

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

   8. Simplified 94.8%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\left|n - m\right| - \color{blue}{\frac{1}{4} \cdot {n}^{2}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto e^{\left|n - m\right| - \color{blue}{\frac{1}{4} \cdot {n}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto e^{\left|n - m\right| - \frac{1}{4} \cdot \color{blue}{\left(n \cdot n\right)}} \]

       3. lower-*.f64 52.8

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

   11. Simplified 52.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -530000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.45 \cdot 10^{-296}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -530000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-194}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -530000000.0)
   (exp (* (* m m) -0.25))
   (if (<= m 2e-194) (exp (* M (- M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -530000000.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= 2e-194) {
		tmp = exp((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 (m <= (-530000000.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= 2d-194) then
        tmp = exp((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 (m <= -530000000.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= 2e-194) {
		tmp = Math.exp((M * -M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -530000000.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= 2e-194:
		tmp = math.exp((M * -M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -530000000.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= 2e-194)
		tmp = exp(Float64(M * Float64(-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 (m <= -530000000.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= 2e-194)
		tmp = exp((M * -M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -530000000.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 2e-194], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.3e8

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 65.0

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

   5. Simplified 65.0%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 98.4

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

   11. Simplified 98.4%

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

   if -5.3e8 < m < 2.00000000000000004e-194

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 92.4

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

   8. Simplified 92.4%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 62.8

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

   11. Simplified 62.8%

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

   if 2.00000000000000004e-194 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 77.0

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

   5. Simplified 77.0%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 94.4

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

   8. Simplified 94.4%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 56.4

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

   11. Simplified 56.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -530000000:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-194}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -110:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -110.0) t_0 (if (<= M 1.2e-11) (exp (* (* m m) -0.25)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -110.0) {
		tmp = t_0;
	} else if (M <= 1.2e-11) {
		tmp = exp(((m * m) * -0.25));
	} 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((m_1 * -m_1))
    if (m_1 <= (-110.0d0)) then
        tmp = t_0
    else if (m_1 <= 1.2d-11) then
        tmp = exp(((m * m) * (-0.25d0)))
    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((M * -M));
	double tmp;
	if (M <= -110.0) {
		tmp = t_0;
	} else if (M <= 1.2e-11) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((M * -M))
	tmp = 0
	if M <= -110.0:
		tmp = t_0
	elif M <= 1.2e-11:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -110.0)
		tmp = t_0;
	elseif (M <= 1.2e-11)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((M * -M));
	tmp = 0.0;
	if (M <= -110.0)
		tmp = t_0;
	elseif (M <= 1.2e-11)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -110.0], t$95$0, If[LessEqual[M, 1.2e-11], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -110 or 1.2000000000000001e-11 < M

   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 M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 77.8

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

   5. Simplified 77.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 96.1

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

   11. Simplified 96.1%

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

   if -110 < M < 1.2000000000000001e-11

   1. Initial program 74.9%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 92.5

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

   8. Simplified 92.5%

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

   9. Taylor expanded in m around inf

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

       1. *-commutative N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 58.8

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

   11. Simplified 58.8%

       \[\leadsto e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -2.75 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 4 \cdot 10^{-14}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -2.75e-5) t_0 (if (<= M 4e-14) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -2.75e-5) {
		tmp = t_0;
	} else if (M <= 4e-14) {
		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((m_1 * -m_1))
    if (m_1 <= (-2.75d-5)) then
        tmp = t_0
    else if (m_1 <= 4d-14) 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((M * -M));
	double tmp;
	if (M <= -2.75e-5) {
		tmp = t_0;
	} else if (M <= 4e-14) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((M * -M))
	tmp = 0
	if M <= -2.75e-5:
		tmp = t_0
	elif M <= 4e-14:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -2.75e-5)
		tmp = t_0;
	elseif (M <= 4e-14)
		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((M * -M));
	tmp = 0.0;
	if (M <= -2.75e-5)
		tmp = t_0;
	elseif (M <= 4e-14)
		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[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -2.75e-5], t$95$0, If[LessEqual[M, 4e-14], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -2.7500000000000001e-5 or 4e-14 < M

   1. Initial program 77.5%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 77.5

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

   5. Simplified 77.5%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.7

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

   8. Simplified 97.7%

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

   9. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 94.7

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

   11. Simplified 94.7%

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

   if -2.7500000000000001e-5 < M < 4e-14

   1. Initial program 75.1%

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

   3. Taylor expanded in M around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 75.1

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

   5. Simplified 75.1%

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

   6. Taylor expanded in K around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. fabs-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 92.4

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

   8. Simplified 92.4%

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

   9. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

       2. lower-neg.f64 46.6

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

   11. Simplified 46.6%

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

Alternative 8: 35.8% 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 76.3%

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

3. Taylor expanded in M around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 76.3

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

5. Simplified 76.3%

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

6. Taylor expanded in K around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. fabs-sub N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. lower-fabs.f64 N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 95.0

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

8. Simplified 95.0%

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

9. Taylor expanded in l around inf

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

    1. neg-mul-1 N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

    2. lower-neg.f64 32.4

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

11. Simplified 32.4%

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

Alternative 9: 7.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in m around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 39.8

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

5. Simplified 39.8%

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

6. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot n\right) - M\right)} \]
7. Step-by-step derivation

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 7.5

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

8. Simplified 7.5%

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

9. Taylor expanded in n around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 7.5

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

11. Simplified 7.5%

    \[\leadsto \color{blue}{\cos M} \]
12. Add Preprocessing

Alternative 10: 7.0% accurate, 359.0× speedup

Math

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

FPCore

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

C

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

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 = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in m around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 39.8

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

5. Simplified 39.8%

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

6. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot n\right) - M\right)} \]
7. Step-by-step derivation

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 7.5

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

8. Simplified 7.5%

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

9. Taylor expanded in n around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 7.5

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

11. Simplified 7.5%

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

12. Taylor expanded in M around 0

    \[\leadsto \color{blue}{1} \]
13. Step-by-step derivation

14. Simplified 7.5%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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)))))))