Average Error: 29.9 → 0.1
Time: 50.3s
Precision: 64
Internal Precision: 4416
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re - im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot (\left(\sqrt{re + im}\right) \cdot \left(\sqrt{re - im}\right) + re)_*}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 29.9

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

  3. Applied difference-of-squares29.9

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

  4. Applied sqrt-prod0.1

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

  5. Applied fma-def0.1

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{(\left(\sqrt{re + im}\right) \cdot \left(\sqrt{re - im}\right) + re)_*}}\]

Runtime

Time bar (total: 50.3s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +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)))))