Average Error: 15.2 → 16.1
Time: 2.6m
Precision: 64
Internal Precision: 1664
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;K \le -9.823664351721304 \cdot 10^{-143}:\\ \;\;\;\;\frac{\cos \left(\left(\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2} \cdot \left(\frac{M}{n} + \frac{M}{m}\right) - K}}{\sqrt[3]{K \cdot M}}\right)}{e^{\left(\ell - \left|m - n\right|\right) + \left(\frac{n + m}{2} - M\right) \cdot \left(\frac{n + m}{2} - M\right)}}\\ \mathbf{if}\;K \le 2.9203578856819853 \cdot 10^{-174}:\\ \;\;\;\;\left(\sqrt{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)} \cdot \sqrt{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left(\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}}\right)\right)}{e^{\left(\ell - \left|m - n\right|\right) + \left(\frac{n + m}{2} - M\right) \cdot \left(\frac{n + m}{2} - M\right)}}\\ \end{array}\]

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 3 regimes

  2. if K < -9.823664351721304e-143

    1. Initial program 20.6

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

    2. Taylor expanded around inf 7.1

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

    3. Applied simplify7.1

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

    5. Applied add-cube-cbrt7.1

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

    7. Applied frac-sub20.4

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

    8. Applied cbrt-div20.4

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

    9. Applied simplify20.5

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

    if -9.823664351721304e-143 < K < 2.9203578856819853e-174

    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-sqr-sqrt20.4

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

    if 2.9203578856819853e-174 < K

    1. Initial program 19.4

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

    2. Taylor expanded around inf 9.5

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

    3. Applied simplify9.5

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

    5. Applied add-cube-cbrt9.4

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

    7. Applied add-cube-cbrt9.5

      \[\leadsto \frac{\cos \left(\left(\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{\frac{1}{2}}{m} + \frac{\frac{1}{2}}{n}}{K} - \frac{1}{M}}}\right)}\right)}{e^{\left(\ell - \left|m - n\right|\right) + \left(\frac{n + m}{2} - M\right) \cdot \left(\frac{n + m}{2} - M\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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)))))))