Average Error: 33.6 → 9.9
Time: 39.3s
Precision: 64
Internal precision: 128
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;im \le -2.422222280270566 \cdot 10^{+26}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re - im\right)} \cdot 0.5\\
\mathbf{if}\;im \le -2.082287675352382 \cdot 10^{-145}:\\
\;\;\;\;\frac{\sqrt{{im}^2 \cdot 2.0}}{\sqrt{\sqrt{{re}^2 + {im}^2} - re}} \cdot 0.5\\
\mathbf{if}\;im \le 3.244880154522944 \cdot 10^{-152}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(2 \cdot re\right)} \cdot 0.5\\
\mathbf{if}\;im \le 4.692828868299432 \cdot 10^{+91}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + {re}^2} + re\right)} \cdot 0.5\\
\mathbf{else}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\
\end{array}\]
Target
| Original | 33.6 |
|---|
| Comparison | 30.2 |
|---|
| Herbie | 9.9 |
|---|
\[ \begin{array}{l}
\mathbf{if}\;re \lt 0:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{{im}^2}{\sqrt{{re}^2 + {im}^2} - re}}\right)\\
\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\
\end{array} \]
Derivation
- Split input into 5 regimes.
-
if im < -2.422222280270566e+26
Initial program 42.4
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Applied simplify 42.4
\[\leadsto \color{blue}{\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + {re}^2} + re\right)} \cdot 0.5}\]
Applied taylor 0.1
\[\leadsto \sqrt{2.0 \cdot \left(re - im\right)} \cdot 0.5\]
Taylor expanded around -inf 0.1
\[\leadsto \sqrt{2.0 \cdot \color{blue}{\left(re - im\right)}} \cdot 0.5\]
if -2.422222280270566e+26 < im < -2.082287675352382e-145
Initial program 24.9
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Applied simplify 24.9
\[\leadsto \color{blue}{\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + {re}^2} + re\right)} \cdot 0.5}\]
- Using strategy
rm
Applied flip-+ 36.6
\[\leadsto \sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{im \cdot im + {re}^2}\right)}^2 - {re}^2}{\sqrt{im \cdot im + {re}^2} - re}}} \cdot 0.5\]
Applied associate-*r/ 36.6
\[\leadsto \sqrt{\color{blue}{\frac{2.0 \cdot \left({\left(\sqrt{im \cdot im + {re}^2}\right)}^2 - {re}^2\right)}{\sqrt{im \cdot im + {re}^2} - re}}} \cdot 0.5\]
Applied sqrt-div 36.7
\[\leadsto \color{blue}{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{im \cdot im + {re}^2}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{im \cdot im + {re}^2} - re}}} \cdot 0.5\]
Applied simplify 26.2
\[\leadsto \frac{\color{blue}{\sqrt{{im}^2 \cdot 2.0}}}{\sqrt{\sqrt{im \cdot im + {re}^2} - re}} \cdot 0.5\]
Applied simplify 26.2
\[\leadsto \frac{\sqrt{{im}^2 \cdot 2.0}}{\color{blue}{\sqrt{\sqrt{{re}^2 + {im}^2} - re}}} \cdot 0.5\]
if -2.082287675352382e-145 < im < 3.244880154522944e-152
Initial program 27.9
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Applied simplify 27.9
\[\leadsto \color{blue}{\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + {re}^2} + re\right)} \cdot 0.5}\]
Applied taylor 0.0
\[\leadsto \sqrt{2.0 \cdot \left(2 \cdot re\right)} \cdot 0.5\]
Taylor expanded around 0 0.0
\[\leadsto \sqrt{2.0 \cdot \color{blue}{\left(2 \cdot re\right)}} \cdot 0.5\]
if 3.244880154522944e-152 < im < 4.692828868299432e+91
Initial program 23.0
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Applied simplify 23.0
\[\leadsto \color{blue}{\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + {re}^2} + re\right)} \cdot 0.5}\]
if 4.692828868299432e+91 < im
Initial program 47.1
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
Applied simplify 47.1
\[\leadsto \color{blue}{\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + {re}^2} + re\right)} \cdot 0.5}\]
Applied taylor 0.2
\[\leadsto \sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\]
Taylor expanded around inf 0.2
\[\leadsto \sqrt{2.0 \cdot \color{blue}{\left(im + re\right)}} \cdot 0.5\]
- Recombined 5 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie --seed '#(3493355826 1170278353 519959925 1517355148 1086412943 3987390220)'
(FPCore (re im)
:name "math.sqrt on complex, real part"
:target
(if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (sqr im) (- (sqrt (+ (sqr re) (sqr im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
(* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
