Average Error: 15.8 → 2.3
Time: 57.8s
Precision: 64
Internal precision: 128
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^2\right) - \left(\ell - \left|m - n\right|\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;K \le -741687408416398.6:\\ \;\;\;\;\left(\cos \left(\frac{\frac{\frac{1}{2}}{m}}{K} - \frac{1}{M}\right) \cdot \cos \left(\frac{\frac{\frac{1}{2}}{K}}{n}\right) - \sin \left(\frac{\frac{\frac{1}{2}}{m}}{K} - \frac{1}{M}\right) \cdot {\left(\sqrt[3]{\sin \left(\frac{\frac{\frac{1}{2}}{K}}{n}\right)}\right)}^3\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^2\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{if}\;K \le 8.475835738159164 \cdot 10^{-56}:\\ \;\;\;\;{\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}\right)}^3 \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^2\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\frac{\frac{1}{2}}{m}}{K} - \frac{1}{M}\right) \cdot \cos \left(\frac{\frac{\frac{1}{2}}{K}}{n}\right) - \sin \left(\frac{\frac{\frac{1}{2}}{m}}{K} - \frac{1}{M}\right) \cdot {\left(\sqrt[3]{\sin \left(\frac{\frac{\frac{1}{2}}{K}}{n}\right)}\right)}^3\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^2\right) - \left(\ell - \left|m - n\right|\right)}\\ \end{array}\]

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 < -741687408416398.6 or 8.475835738159164e-56 < K

    1. Initial program 29.8

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

    2. Applied taylor 4.3

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

    3. Taylor expanded around inf 4.3

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

    5. Applied associate--l+ 4.3

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

    6. Applied cos-sum 4.2

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

    7. Applied simplify 4.2

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

    8. Applied simplify 4.2

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

    10. Applied add-cube-cbrt 4.2

      \[\leadsto \left(\cos \left(\frac{\frac{\frac{1}{2}}{m}}{K} - \frac{1}{M}\right) \cdot \cos \left(\frac{\frac{\frac{1}{2}}{K}}{n}\right) - \sin \left(\frac{\frac{\frac{1}{2}}{m}}{K} - \frac{1}{M}\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(\frac{\frac{\frac{1}{2}}{K}}{n}\right)}\right)}^3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^2\right) - \left(\ell - \left|m - n\right|\right)}\]

    if -741687408416398.6 < K < 8.475835738159164e-56

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt 0.0

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

Runtime

Time bar (total: 57.8s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(865073390 1317354639 1085565020 880982919 61758579 2507326756)'
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (sqr (- (/ (+ m n) 2) M))) (- l (fabs (- m n)))))))