Average Error: 15.2 → 1.7
Time: 4.7m
Precision: 64
Internal Precision: 1408
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{K \cdot \left(m + n\right)}{2} - M \le -inf.0:\\ \;\;\;\;\frac{\cos \left(\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}\right)}{e^{\left(\ell - \left|m - n\right|\right) + \left(\frac{n + m}{2} - M\right) \cdot \left(\frac{n + m}{2} - M\right)}}\\ \mathbf{if}\;\frac{K \cdot \left(m + n\right)}{2} - M \le 3.750861873783077 \cdot 10^{+279}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)} \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}}\right)\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}\right)}{e^{\left(\ell - \left|m - n\right|\right) + \left(\frac{n + m}{2} - M\right) \cdot \left(\frac{n + m}{2} - M\right)}}\\ \end{array}\]

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (- (/ (* K (+ m n)) 2) M) or 3.750861873783077e+279 < (- (/ (* K (+ m n)) 2) M)

    1. Initial program 53.9

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

    2. Taylor expanded around inf 1.3

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

    3. Applied simplify1.3

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

    if (- (/ (* K (+ m n)) 2) M) < 3.750861873783077e+279

    1. Initial program 1.8

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

    3. Applied add-cube-cbrt1.8

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

    5. Applied add-cube-cbrt1.8

      \[\leadsto \left(\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)} \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}}\right)}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 4.7m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))