Average Error: 38.9 → 13.4
Time: 3.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -1.47033881827273 \cdot 10^{-87} \lor \neg \left(im \le -1.1781651487664911 \cdot 10^{-125} \lor \neg \left(im \le 1.0908647843737489 \cdot 10^{-38} \lor \neg \left(im \le 6.702754729213891 \cdot 10^{102}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -1.47033881827273 \cdot 10^{-87} \lor \neg \left(im \le -1.1781651487664911 \cdot 10^{-125} \lor \neg \left(im \le 1.0908647843737489 \cdot 10^{-38} \lor \neg \left(im \le 6.702754729213891 \cdot 10^{102}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

\end{array}
double f(double re, double im) {
        double r318266 = 0.5;
        double r318267 = 2.0;
        double r318268 = re;
        double r318269 = r318268 * r318268;
        double r318270 = im;
        double r318271 = r318270 * r318270;
        double r318272 = r318269 + r318271;
        double r318273 = sqrt(r318272);
        double r318274 = r318273 + r318268;
        double r318275 = r318267 * r318274;
        double r318276 = sqrt(r318275);
        double r318277 = r318266 * r318276;
        return r318277;
}
↓
double f(double re, double im) {
        double r318278 = im;
        double r318279 = -1.47033881827273e-87;
        bool r318280 = r318278 <= r318279;
        double r318281 = -1.1781651487664911e-125;
        bool r318282 = r318278 <= r318281;
        double r318283 = 1.090864784373749e-38;
        bool r318284 = r318278 <= r318283;
        double r318285 = 6.702754729213891e+102;
        bool r318286 = r318278 <= r318285;
        double r318287 = !r318286;
        bool r318288 = r318284 || r318287;
        double r318289 = !r318288;
        bool r318290 = r318282 || r318289;
        double r318291 = !r318290;
        bool r318292 = r318280 || r318291;
        double r318293 = 0.5;
        double r318294 = 2.0;
        double r318295 = 1.0;
        double r318296 = re;
        double r318297 = hypot(r318296, r318278);
        double r318298 = r318295 * r318297;
        double r318299 = r318298 + r318296;
        double r318300 = r318294 * r318299;
        double r318301 = sqrt(r318300);
        double r318302 = r318293 * r318301;
        double r318303 = 2.0;
        double r318304 = pow(r318278, r318303);
        double r318305 = r318297 - r318296;
        double r318306 = r318304 / r318305;
        double r318307 = r318294 * r318306;
        double r318308 = sqrt(r318307);
        double r318309 = r318293 * r318308;
        double r318310 = r318292 ? r318302 : r318309;
        return r318310;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.9
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < -1.47033881827273e-87 or -1.1781651487664911e-125 < im < 1.090864784373749e-38 or 6.702754729213891e+102 < im

    1. Initial program 41.5

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

    3. Applied *-un-lft-identity41.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}} + re\right)}\]

    4. Applied sqrt-prod41.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}} + re\right)}\]

    5. Simplified41.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im} + re\right)}\]

    6. Simplified12.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]

    if -1.47033881827273e-87 < im < -1.1781651487664911e-125 or 1.090864784373749e-38 < im < 6.702754729213891e+102

    1. Initial program 23.2

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

    3. Applied flip-+30.3

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

    4. Simplified23.1

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

    5. Simplified17.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.47033881827273 \cdot 10^{-87} \lor \neg \left(im \le -1.1781651487664911 \cdot 10^{-125} \lor \neg \left(im \le 1.0908647843737489 \cdot 10^{-38} \lor \neg \left(im \le 6.702754729213891 \cdot 10^{102}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))