Average Error: 37.9 → 0.6
Time: 1.0m
Precision: 64
Internal Precision: 3392
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \le 2.276927232505162 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{-\frac{1.0}{re}}}{\left|\frac{-1}{im}\right|} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original37.9
Target32.8
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* 0.5 (sqrt (fma (hypot re im) 2.0 (* re 2.0)))) < 2.276927232505162e-131

    1. Initial program 60.0

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

    2. Applied simplify53.0

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

    4. Applied add-exp-log53.1

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

    5. Taylor expanded around -inf 44.1

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

    6. Applied simplify22.8

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

    8. Applied frac-times29.9

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

    9. Applied sqrt-div22.6

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

    10. Applied simplify22.6

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

    11. Applied simplify1.4

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

    if 2.276927232505162e-131 < (* 0.5 (sqrt (fma (hypot re im) 2.0 (* re 2.0))))

    1. Initial program 30.3

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

    2. Applied simplify0.3

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018214 +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)))))