Average Error: 38.9 → 27.0
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 8.5549477735948 \cdot 10^{74}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 8.5549477735948 \cdot 10^{74}:\\
\;\;\;\;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(re + im\right)}\\

\end{array}
double f(double re, double im) {
        double r141204 = 0.5;
        double r141205 = 2.0;
        double r141206 = re;
        double r141207 = r141206 * r141206;
        double r141208 = im;
        double r141209 = r141208 * r141208;
        double r141210 = r141207 + r141209;
        double r141211 = sqrt(r141210);
        double r141212 = r141211 + r141206;
        double r141213 = r141205 * r141212;
        double r141214 = sqrt(r141213);
        double r141215 = r141204 * r141214;
        return r141215;
}
↓
double f(double re, double im) {
        double r141216 = 2.0;
        double r141217 = re;
        double r141218 = r141217 * r141217;
        double r141219 = im;
        double r141220 = r141219 * r141219;
        double r141221 = r141218 + r141220;
        double r141222 = sqrt(r141221);
        double r141223 = r141222 + r141217;
        double r141224 = r141216 * r141223;
        double r141225 = sqrt(r141224);
        double r141226 = 0.0;
        bool r141227 = r141225 <= r141226;
        double r141228 = 0.5;
        double r141229 = 2.0;
        double r141230 = pow(r141219, r141229);
        double r141231 = r141222 - r141217;
        double r141232 = r141230 / r141231;
        double r141233 = r141216 * r141232;
        double r141234 = sqrt(r141233);
        double r141235 = r141228 * r141234;
        double r141236 = 8.5549477735948e+74;
        bool r141237 = r141225 <= r141236;
        double r141238 = r141228 * r141225;
        double r141239 = r141217 + r141219;
        double r141240 = r141216 * r141239;
        double r141241 = sqrt(r141240);
        double r141242 = r141228 * r141241;
        double r141243 = r141237 ? r141238 : r141242;
        double r141244 = r141227 ? r141235 : r141243;
        return r141244;
}

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
Herbie27.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 0.0

    1. Initial program 57.2

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

    3. Applied add-sqr-sqrt57.2

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

    4. Applied sqrt-prod59.6

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

    6. Applied sqrt-unprod57.2

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

    7. Simplified57.2

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

    9. Applied flip-+57.2

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

    10. Simplified29.1

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

    if 0.0 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 8.5549477735948e+74

    1. Initial program 4.4

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

    3. Applied add-sqr-sqrt4.4

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

    4. Applied sqrt-prod4.5

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

    6. Applied sqrt-unprod4.4

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

    7. Simplified4.4

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

    if 8.5549477735948e+74 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))

    1. Initial program 63.0

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

    2. Taylor expanded around 0 45.3

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

  4. Final simplification27.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 8.5549477735948 \cdot 10^{74}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(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)))))