Average Error: 2.1 → 2.2
Time: 9.4s
Precision: 64
Internal Precision: 320
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \sqrt{re \cdot re + im \cdot im} + 2.0 \cdot re}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

    \[\left(real->posit(0.5)\right) \cdot \left(\sqrt{\left(\left(real->posit(2.0)\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in2.2

    \[\leadsto \left(real->posit(0.5)\right) \cdot \left(\sqrt{\color{blue}{\left(\frac{\left(\left(real->posit(2.0)\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right)}{\left(\left(real->posit(2.0)\right) \cdot re\right)}\right)}}\right)\]

  4. Final simplification2.2

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \sqrt{re \cdot re + im \cdot im} + 2.0 \cdot re}\]

Reproduce

herbie shell --seed 2019090 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (+.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))