Average Error: 47.4 → 28.4
Time: 7.7m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.861181919236283 \cdot 10^{-191}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \left(\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\right)}\\ \mathbf{if}\;t \le 2.66152726250732 \cdot 10^{-220}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{1}{{\ell}^{2} \cdot {t}^{3}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \left(\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.861181919236283e-191 or 2.66152726250732e-220 < t

    1. Initial program 45.1

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt45.2

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}}\]

    4. Applied associate-*r*45.2

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}}\]

    5. Applied simplify38.0

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right)} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt38.0

      \[\leadsto \frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \cdot \sqrt{\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\right)}}\]

    8. Applied simplify38.0

      \[\leadsto \frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \left(\color{blue}{\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}} \cdot \sqrt{\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\right)}\]

    9. Applied simplify26.2

      \[\leadsto \frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \left(\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \color{blue}{\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}\right)}\]

    if -1.861181919236283e-191 < t < 2.66152726250732e-220

    1. Initial program 62.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

    2. Taylor expanded around inf 43.3

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{1}{{\ell}^{2} \cdot {t}^{3}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 7.7m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))