Average Error: 15.2 → 1.5
Time: 2.1m
Precision: 64
Internal Precision: 1664
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{K \cdot \left(m + n\right)}{2} - M \le -inf.0:\\ \;\;\;\;\left(\cos \left(\frac{\frac{1}{2}}{m \cdot K}\right) \cdot \cos \left(\frac{1}{M}\right) - \sin \left(\frac{\frac{\frac{1}{2}}{m}}{K}\right) \cdot \left(-\sin \left(\frac{1}{M}\right)\right)\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{if}\;\frac{K \cdot \left(m + n\right)}{2} - M \le 1.2222768544040524 \cdot 10^{+286}:\\ \;\;\;\;\sqrt[3]{{\left(\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)\right)}^{3}} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\frac{1}{2}}{m \cdot K}\right) \cdot \cos \left(\frac{1}{M}\right) - \sin \left(\frac{\frac{\frac{1}{2}}{m}}{K}\right) \cdot \left(-\sin \left(\frac{1}{M}\right)\right)\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\\ \end{array}\]

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) or 1.2222768544040524e+286 < (- (/ (* K (+ m n)) 2) M)

    1. Initial program 56.0

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

    3. Applied add-cube-cbrt56.0

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

    4. Applied associate-*r*56.0

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

    5. Taylor expanded around inf 0.9

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

    7. Applied sub-neg0.9

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

    8. Applied cos-sum0.9

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

    9. Applied simplify0.9

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

    10. Applied simplify0.9

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

    if (- (/ (* K (+ m n)) 2) M) < 1.2222768544040524e+286

    1. Initial program 1.7

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

    3. Applied add-cbrt-cube1.7

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

    4. Applied simplify1.7

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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)))))))