Average Error: 37.8 → 13.7

Time: 10.5s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Initial program 37.8
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
Simplified13.7
\[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0}}\]
Final simplification13.7
\[\leadsto \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019030 +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)))))