Average Error: 37.6 → 13.7

Time: 11.0s

Precision: 64

Internal Precision: 3648

\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

↓

\[\left(\left(\sqrt{\sqrt{\sqrt{re^2 + im^2}^* - re}} \cdot \sqrt{\sqrt{\sqrt{re^2 + im^2}^* - re}}\right) \cdot 0.5\right) \cdot \sqrt{2.0}\]

Error

The X axis uses an exponential scale — Bits error versus `re`

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Derivation

Initial program 37.6
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
Initial simplification13.2
\[\leadsto 0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0}\]
Using strategy rm
Applied sqrt-prod13.5
\[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{re^2 + im^2}^* - re} \cdot \sqrt{2.0}\right)}\]
Applied associate-*r*13.5
\[\leadsto \color{blue}{\left(0.5 \cdot \sqrt{\sqrt{re^2 + im^2}^* - re}\right) \cdot \sqrt{2.0}}\]
Using strategy rm
Applied add-sqr-sqrt13.7
\[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt{re^2 + im^2}^* - re}} \cdot \sqrt{\sqrt{\sqrt{re^2 + im^2}^* - re}}\right)}\right) \cdot \sqrt{2.0}\]
Final simplification13.7
\[\leadsto \left(\left(\sqrt{\sqrt{\sqrt{re^2 + im^2}^* - re}} \cdot \sqrt{\sqrt{\sqrt{re^2 + im^2}^* - re}}\right) \cdot 0.5\right) \cdot \sqrt{2.0}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	13.7	13.7	13.0	0.7	0%

herbie shell --seed 2018297 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))