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Approximations for hand calculators using small integer coefficients

Author: Stephen E. Derenzo
Journal: Math. Comp. 31 (1977), 214-222
MSC: Primary 65D20
MathSciNet review: 0423761
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Abstract: Methods are presented for deriving approximations containing small integer coefficients. This approach is useful for electronic hand calculators and programmable calculators, where it is important to minimize the number of keystrokes necessary to evaluate the function. For example, the probability $ P(x)$ of exceeding x standard deviations of either sign (Gaussian probability integral) is approximated by

$\displaystyle P(x) \approx {\text{EXP}}\left[ { - \frac{{(83x + 351)x + 562}}{{703/x + 165}}} \right]$

with a relative error less than 0.042

References [Enhancements On Off] (What's this?)

  • [1] C. HASTINGS, JR., Approximations for Digital Computers, Princeton Univ. Press, Princeton, N. J., 1955. MR 16, 963. MR 0068915 (16:963e)
  • [2] J. F. HART ET AL., Computer Approximations, Wiley, New York, 1968.
  • [3] For cases where the absolute error $ g(x) - f(x)$ is important, $ w(x)$ is equal to a constant. For cases where the relative error $ (g(x) - f(x))/f(x)$ is important, $ w(x)$ is proportional to $ 1/f(x)$.
  • [4] STEPHEN E. DERENZO, Lawrence Radiation Laboratory Group A Programming Note P-190, Berkeley, Calif., 1969. Most other minimizing codes can also be used.
  • [5] STEPHEN E. DERENZO, Lawrence Berkeley Laboratory Report No. LBL-3804, Berkeley, Calif., March, 1975. (Available from the author.)
  • [6] Henceforth we use the term "best fit coefficients" to mean those that result from the minimization of D (Eq. (2)).
  • [7] The number given after the $ \pm $ symbol is the amount that the corresponding coefficient must be varied from its best fit value to double the value of D, holding all other coefficients at their best fit values.
  • [8] O. KLEIN AND Y. NISHINA, Nature, v. 122, 1928, p. 398. Formulas are also available in the American Institute of Physics Handbook, 3rd ed. (D. E. Gray, Editor), McGraw-Hill, New York, 1972, pp. 8-197.
  • [9] J. H. HUBBELL, Photon Cross Sections, Attenuation Coefficients, and Energy Absorption Coefficients from 10 keV to 100 GeV, Report No. NSRDS-NBS 29, U. S. Nat. Bur. of Standards, 1969.

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Article copyright: © Copyright 1977 American Mathematical Society

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