Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 97.1%

Time: 12.7s

Alternatives: 9

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
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 98.3

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 2: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -0.175:\\ \;\;\;\;\cos M \cdot t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -0.175)
     (* (cos M) t_0)
     (if (<= M 27.0)
       (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 <= -0.175) {
		tmp = cos(M) * t_0;
	} else if (M <= 27.0) {
		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 <= -0.175)
		tmp = Float64(cos(M) * t_0);
	elseif (M <= 27.0)
		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, -0.175], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 27.0], 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 3 regimes

2. if M < -0.17499999999999999

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 95.5

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

   8. Simplified 95.5%

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

   if -0.17499999999999999 < M < 27

   1. Initial program 71.8%

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

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

   if 27 < M

   1. Initial program 75.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in M around 0

      \[\leadsto \color{blue}{1} \cdot e^{M \cdot \left(\mathsf{neg}\left(M\right)\right)} \]
   10. Step-by-step derivation

   11. Simplified 100.0%

       \[\leadsto \color{blue}{1} \cdot e^{M \cdot \left(-M\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -0.175:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -0.175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -0.175)
     t_0
     (if (<= M 27.0)
       (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 <= -0.175) {
		tmp = t_0;
	} else if (M <= 27.0) {
		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 <= -0.175)
		tmp = t_0;
	elseif (M <= 27.0)
		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, -0.175], t$95$0, If[LessEqual[M, 27.0], 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 < -0.17499999999999999 or 27 < M

   1. Initial program 77.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 98.6

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

   5. Simplified 98.6%

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

   6. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 97.7

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

   8. Simplified 97.7%

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

   9. Taylor expanded in M around 0

      \[\leadsto \color{blue}{1} \cdot e^{M \cdot \left(\mathsf{neg}\left(M\right)\right)} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

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

   if -0.17499999999999999 < M < 27

   1. Initial program 71.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -0.175:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.105:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.6 \cdot 10^{-288}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.105)
   (exp (* (* m m) -0.25))
   (if (<= m -2.6e-288) (exp (* M (- M))) (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.105) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -2.6e-288) {
		tmp = exp((M * -M));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-0.105d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-2.6d-288)) then
        tmp = exp((m_1 * -m_1))
    else
        tmp = exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.105) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -2.6e-288) {
		tmp = Math.exp((M * -M));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.105:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -2.6e-288:
		tmp = math.exp((M * -M))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.105)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -2.6e-288)
		tmp = exp(Float64(M * Float64(-M)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.105)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -2.6e-288)
		tmp = exp((M * -M));
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.105], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -2.6e-288], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.104999999999999996

   1. Initial program 70.2%

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

   3. Taylor expanded in 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 99.9

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

   5. Simplified 99.9%

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

   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 98.6

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

   8. Simplified 98.6%

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

   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 96.1

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

   11. Simplified 96.1%

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

   if -0.104999999999999996 < m < -2.59999999999999989e-288

   1. Initial program 81.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 61.2

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

   8. Simplified 61.2%

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

   9. Taylor expanded in M around 0

      \[\leadsto \color{blue}{1} \cdot e^{M \cdot \left(\mathsf{neg}\left(M\right)\right)} \]
   10. Step-by-step derivation

   11. Simplified 61.2%

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

   if -2.59999999999999989e-288 < m

   1. Initial program 73.6%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.5

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

   5. Simplified 97.5%

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

   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 87.4

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

   8. Simplified 87.4%

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

   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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 60.7

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

   11. Simplified 60.7%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.105:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.6 \cdot 10^{-288}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -42:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.15 \cdot 10^{-157}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -42.0)
   (exp (* (* m m) -0.25))
   (if (<= m -1.15e-157) (exp (- l)) (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -42.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -1.15e-157) {
		tmp = exp(-l);
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-42.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-1.15d-157)) then
        tmp = exp(-l)
    else
        tmp = exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -42.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -1.15e-157) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -42.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -1.15e-157:
		tmp = math.exp(-l)
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -42.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -1.15e-157)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -42.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -1.15e-157)
		tmp = exp(-l);
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -42.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1.15e-157], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -42

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 99.9

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

   5. Simplified 99.9%

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

   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 98.5

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

   8. Simplified 98.5%

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

   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 97.4

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

   11. Simplified 97.4%

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

   if -42 < m < -1.14999999999999994e-157

   1. Initial program 90.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 100.0

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

   5. Simplified 100.0%

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

   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 80.5

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

   8. Simplified 80.5%

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

   9. Taylor expanded in l around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 54.6

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

   11. Simplified 54.6%

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

   if -1.14999999999999994e-157 < m

   1. Initial program 73.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.3

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

   5. Simplified 97.3%

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

   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 83.8

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

   8. Simplified 83.8%

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

   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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 59.8

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

   11. Simplified 59.8%

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

Alternative 6: 68.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{n \cdot \left(n \cdot -0.25\right)}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* n (* n -0.25)))))
   (if (<= n -2.9e+19) t_0 (if (<= n 4.3e-14) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((n * (n * -0.25)));
	double tmp;
	if (n <= -2.9e+19) {
		tmp = t_0;
	} else if (n <= 4.3e-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((n * (n * (-0.25d0))))
    if (n <= (-2.9d+19)) then
        tmp = t_0
    else if (n <= 4.3d-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((n * (n * -0.25)));
	double tmp;
	if (n <= -2.9e+19) {
		tmp = t_0;
	} else if (n <= 4.3e-14) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((n * (n * -0.25)))
	tmp = 0
	if n <= -2.9e+19:
		tmp = t_0
	elif n <= 4.3e-14:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(n * Float64(n * -0.25)))
	tmp = 0.0
	if (n <= -2.9e+19)
		tmp = t_0;
	elseif (n <= 4.3e-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((n * (n * -0.25)));
	tmp = 0.0;
	if (n <= -2.9e+19)
		tmp = t_0;
	elseif (n <= 4.3e-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[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -2.9e+19], t$95$0, If[LessEqual[n, 4.3e-14], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9e19 or 4.29999999999999998e-14 < n

   1. Initial program 68.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 99.2

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

   5. Simplified 99.2%

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

   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 96.2

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

   8. Simplified 96.2%

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

   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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 94.0

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

   11. Simplified 94.0%

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

   if -2.9e19 < n < 4.29999999999999998e-14

   1. Initial program 80.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 97.4

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

   5. Simplified 97.4%

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

   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 78.5

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

   8. Simplified 78.5%

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

   9. Taylor expanded in l around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 41.3

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

   11. Simplified 41.3%

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

Alternative 7: 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 74.4%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
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 98.3

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

5. Simplified 98.3%

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

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 87.6

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

8. Simplified 87.6%

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

9. Taylor expanded in l around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 35.6

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

11. Simplified 35.6%

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

Alternative 8: 7.3% 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 74.4%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 40.4

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

5. Simplified 40.4%

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

6. Taylor expanded in n around 0

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

   1. lower-cos.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 8.1

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

8. Simplified 8.1%

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

9. Taylor expanded in K 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 8.6

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

11. Simplified 8.6%

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

Alternative 9: 7.3% 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 74.4%

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

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)} \]
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 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in M around inf

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 49.9

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

8. Simplified 49.9%

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

9. Taylor expanded in M around 0

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

11. Simplified 8.6%

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

Reproduce

herbie shell --seed 2024207 
(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)))))))