Maksimov and Kolovsky, Equation (32)

Average Error: 14.5 → 1.4

Time: 7.9s

Precision: binary64

\[ \begin{array}{c}[m, n] = \mathsf{sort}([m, n])\\ \end{array} \]

Math

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

↓

\[\log \left({\left(e^{\cos M}\right)}^{\left(e^{\left(n - m\right) - \left({\left(\left(n + m\right) \cdot 0.5 - M\right)}^{2} + \ell\right)}\right)}\right) \]

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)))))))

↓

(FPCore (K m n M l)
 :precision binary64
 (log
  (pow (exp (cos M)) (exp (- (- n m) (+ (pow (- (* (+ n m) 0.5) M) 2.0) l))))))

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)))));
}

↓

double code(double K, double m, double n, double M, double l) {
	return log(pow(exp(cos(M)), exp(((n - m) - (pow((((n + m) * 0.5) - M), 2.0) + l)))));
}

Fortran

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

↓

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 = log((exp(cos(m_1)) ** exp(((n - m) - (((((n + m) * 0.5d0) - m_1) ** 2.0d0) + l)))))
end function

Java

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

↓

public static double code(double K, double m, double n, double M, double l) {
	return Math.log(Math.pow(Math.exp(Math.cos(M)), Math.exp(((n - m) - (Math.pow((((n + m) * 0.5) - M), 2.0) + l)))));
}

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)))))

↓

def code(K, m, n, M, l):
	return math.log(math.pow(math.exp(math.cos(M)), math.exp(((n - m) - (math.pow((((n + m) * 0.5) - M), 2.0) + l)))))

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

↓

function code(K, m, n, M, l)
	return log((exp(cos(M)) ^ exp(Float64(Float64(n - m) - Float64((Float64(Float64(Float64(n + m) * 0.5) - M) ^ 2.0) + l)))))
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

↓

function tmp = code(K, m, n, M, l)
	tmp = log((exp(cos(M)) ^ exp(((n - m) - (((((n + m) * 0.5) - M) ^ 2.0) + l)))));
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]

↓

code[K_, m_, n_, M_, l_] := N[Log[N[Power[N[Exp[N[Cos[M], $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n - m), $MachinePrecision] - N[(N[Power[N[(N[(N[(n + m), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Derivation

1. Initial program 14.5

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

2. Simplified 14.5

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

3. Taylor expanded in K around 0 1.3

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

4. Simplified 1.3

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

5. Applied egg-rr 1.4

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

6. Final simplification 1.4

   \[\leadsto \log \left({\left(e^{\cos M}\right)}^{\left(e^{\left(n - m\right) - \left({\left(\left(n + m\right) \cdot 0.5 - M\right)}^{2} + \ell\right)}\right)}\right) \]

Reproduce

herbie shell --seed 2022165 
(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)))))))