Average Error: 37.5 → 7.4
Time: 27.9s
Precision: 64
Internal Precision: 2368
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5497065.5798862465:\\ \;\;\;\;\left(\sqrt{\frac{-1.0}{re}} \cdot \left|im\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot 0.5\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original37.5
Target32.6
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -5497065.5798862465

    1. Initial program 55.7

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

    2. Initial simplification38.6

      \[\leadsto 0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\]
    3. Using strategy rm

    4. Applied add-exp-log39.9

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left((\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*\right)}}}\]

    5. Taylor expanded around -inf 46.6

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{e^{\left(\log \left(\frac{-1}{re}\right) + \log 1.0\right) - 2 \cdot \log \left(\frac{-1}{im}\right)}}}\]

    6. Simplified29.1

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}}}\]
    7. Using strategy rm

    8. Applied frac-times32.4

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{\frac{-1}{re} \cdot 1.0}{\frac{-1}{im} \cdot \frac{-1}{im}}}}\]

    9. Applied sqrt-div25.5

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{\frac{-1}{re} \cdot 1.0}}{\sqrt{\frac{-1}{im} \cdot \frac{-1}{im}}}}\]

    10. Simplified25.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-\frac{1.0}{re}}}}{\sqrt{\frac{-1}{im} \cdot \frac{-1}{im}}}\]
    11. Using strategy rm

    12. Applied frac-times25.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{-\frac{1.0}{re}}}{\sqrt{\color{blue}{\frac{-1 \cdot -1}{im \cdot im}}}}\]

    13. Applied sqrt-div25.4

      \[\leadsto 0.5 \cdot \frac{\sqrt{-\frac{1.0}{re}}}{\color{blue}{\frac{\sqrt{-1 \cdot -1}}{\sqrt{im \cdot im}}}}\]

    14. Applied associate-/r/25.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{-\frac{1.0}{re}}}{\sqrt{-1 \cdot -1}} \cdot \sqrt{im \cdot im}\right)}\]

    15. Simplified14.0

      \[\leadsto 0.5 \cdot \left(\frac{\sqrt{-\frac{1.0}{re}}}{\sqrt{-1 \cdot -1}} \cdot \color{blue}{\left|im\right|}\right)\]

    if -5497065.5798862465 < re

    1. Initial program 31.9

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

    2. Initial simplification5.4

      \[\leadsto 0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5497065.5798862465:\\ \;\;\;\;\left(\sqrt{\frac{-1.0}{re}} \cdot \left|im\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot 0.5\\ \end{array}\]

Runtime

Time bar (total: 27.9s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))