Average Error: 39.2 → 27.7
Time: 4.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 2.74236394744986865 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 8.74783628294273192 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \le 0.0:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 2.74236394744986865 \cdot 10^{-181}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 8.74783628294273192 \cdot 10^{153}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\end{array}
double f(double re, double im) {
        double r19226 = 0.5;
        double r19227 = 2.0;
        double r19228 = re;
        double r19229 = r19228 * r19228;
        double r19230 = im;
        double r19231 = r19230 * r19230;
        double r19232 = r19229 + r19231;
        double r19233 = sqrt(r19232);
        double r19234 = r19233 - r19228;
        double r19235 = r19227 * r19234;
        double r19236 = sqrt(r19235);
        double r19237 = r19226 * r19236;
        return r19237;
}
↓
double f(double re, double im) {
        double r19238 = re;
        double r19239 = r19238 * r19238;
        double r19240 = im;
        double r19241 = r19240 * r19240;
        double r19242 = r19239 + r19241;
        double r19243 = sqrt(r19242);
        double r19244 = r19243 - r19238;
        double r19245 = 0.0;
        bool r19246 = r19244 <= r19245;
        double r19247 = 0.5;
        double r19248 = 2.0;
        double r19249 = 2.0;
        double r19250 = pow(r19240, r19249);
        double r19251 = r19243 + r19238;
        double r19252 = r19250 / r19251;
        double r19253 = r19248 * r19252;
        double r19254 = sqrt(r19253);
        double r19255 = r19247 * r19254;
        double r19256 = 2.7423639474498686e-181;
        bool r19257 = r19244 <= r19256;
        double r19258 = -2.0;
        double r19259 = r19258 * r19238;
        double r19260 = r19248 * r19259;
        double r19261 = sqrt(r19260);
        double r19262 = r19247 * r19261;
        double r19263 = 8.747836282942732e+153;
        bool r19264 = r19244 <= r19263;
        double r19265 = r19248 * r19244;
        double r19266 = sqrt(r19265);
        double r19267 = r19247 * r19266;
        double r19268 = r19264 ? r19267 : r19262;
        double r19269 = r19257 ? r19262 : r19268;
        double r19270 = r19246 ? r19255 : r19269;
        return r19270;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (sqrt (+ (* re re) (* im im))) re) < 0.0

    1. Initial program 58.4

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

    3. Applied flip--57.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Simplified37.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    if 0.0 < (- (sqrt (+ (* re re) (* im im))) re) < 2.7423639474498686e-181 or 8.747836282942732e+153 < (- (sqrt (+ (* re re) (* im im))) re)

    1. Initial program 63.4

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

    2. Taylor expanded around -inf 45.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if 2.7423639474498686e-181 < (- (sqrt (+ (* re re) (* im im))) re) < 8.747836282942732e+153

    1. Initial program 1.1

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification27.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 2.74236394744986865 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 8.74783628294273192 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))