Average Error: 15.1 → 1.7
Time: 2.2m
Precision: 64
Internal Precision: 1408
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{K \cdot \left(m + n\right)}{2} - M \le -3.653191116193967 \cdot 10^{+305}:\\ \;\;\;\;\cos \left(\left(\frac{1}{2} \cdot \frac{1}{m \cdot K} + \frac{1}{2} \cdot \frac{1}{n \cdot K}\right) - \frac{1}{M}\right) \cdot {e}^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}\\ \mathbf{if}\;\frac{K \cdot \left(m + n\right)}{2} - M \le 4.0009193320351597 \cdot 10^{+279}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)} \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}}\right)\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\frac{1}{2} \cdot \frac{1}{m \cdot K} + \frac{1}{2} \cdot \frac{1}{n \cdot K}\right) - \frac{1}{M}\right) \cdot {e}^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}\\ \end{array}\]

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) < -3.653191116193967e+305 or 4.0009193320351597e+279 < (- (/ (* K (+ m n)) 2) M)

    1. Initial program 52.7

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

    3. Applied *-un-lft-identity52.7

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

    4. Applied exp-prod52.7

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

    5. Applied simplify52.7

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

    6. Taylor expanded around inf 1.6

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

    if -3.653191116193967e+305 < (- (/ (* K (+ m n)) 2) M) < 4.0009193320351597e+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: 2.2m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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)))))))