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Jan 2, 2005 - The efficient computation of transcendental functions for complex floating point numbers is an important i
Mathematics: Complex functions Marcel Leutenegger ´Ecole Polytechnique F´ed´erale de Lausanne Laboratoire d’Optique Biom´edicale 1015 Lausanne, Switzerland January 2, 2005 The efficient computation of transcendental functions for complex floating point numbers is an important issue for any number crunching software as MATLAB for example. Efficient code is based on the analytical simplification of the function definition together with a reuse of intermediate values. The limited precision of the representation of floating point values can lead to significant errors. In particular, the evaluation of complex values frequently involves additions and subtractions. Their round-off error potentially degrades the accuracy of the final result. This issue is briefly discussed within this document.
1 Introduction Complex values z are represented as two floating point numbers representing the real and the imaginary part respectively. Hence √ z = 29 b2
(3)
∀ b2 > 29 a2
(4)
The ratio is 2 to the power of the mantissa length difference (64bit - 53bit) less 2 rounding bits, hence 29 = 512. The logarithm involved in the inverse transcendental functions is always computed with equation (2). Due to the three to four addends involved, split-up leads to a manifold of different expressions. Conditional branches to these expressions are unpredictable and degrade the overall performance significantly. Argument
Result
−∞
+∞+iπ
n
r+iπ −∞
0 p
r
+∞
+∞
r+ir
r+i f
3
2.3 Square root
2 BASIC FUNCTIONS
2.3 Square root In principle, the square root could be computed as r r |z| + a |z| − a sqrt (z) = ±i 2 2
|z| =
with
p
a2 + b2
(5)
But for |a| |b|, the imaginary part would be truncated to zero. To maintain precision over the entire complex plane R × iR, the square root is implemented as r
r |z| + a 2 b sqrt (z) = +i 2 2 |z| + a r r 2 |z| − a |b| ±i sqrt (z) = 2 |z| − a 2
