Average Error: 39.4 → 27.1
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 3.487300811359298841390450553085918164253 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]
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 3.487300811359298841390450553085918164253 \cdot 10^{76}:\\
\;\;\;\;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 r136960 = 0.5;
        double r136961 = 2.0;
        double r136962 = re;
        double r136963 = r136962 * r136962;
        double r136964 = im;
        double r136965 = r136964 * r136964;
        double r136966 = r136963 + r136965;
        double r136967 = sqrt(r136966);
        double r136968 = r136967 + r136962;
        double r136969 = r136961 * r136968;
        double r136970 = sqrt(r136969);
        double r136971 = r136960 * r136970;
        return r136971;
}
↓
double f(double re, double im) {
        double r136972 = 2.0;
        double r136973 = re;
        double r136974 = r136973 * r136973;
        double r136975 = im;
        double r136976 = r136975 * r136975;
        double r136977 = r136974 + r136976;
        double r136978 = sqrt(r136977);
        double r136979 = r136978 + r136973;
        double r136980 = r136972 * r136979;
        double r136981 = sqrt(r136980);
        double r136982 = 0.0;
        bool r136983 = r136981 <= r136982;
        double r136984 = 0.5;
        double r136985 = 2.0;
        double r136986 = pow(r136975, r136985);
        double r136987 = r136978 - r136973;
        double r136988 = r136986 / r136987;
        double r136989 = r136972 * r136988;
        double r136990 = sqrt(r136989);
        double r136991 = r136984 * r136990;
        double r136992 = 3.487300811359299e+76;
        bool r136993 = r136981 <= r136992;
        double r136994 = r136984 * r136981;
        double r136995 = r136973 + r136975;
        double r136996 = r136972 * r136995;
        double r136997 = sqrt(r136996);
        double r136998 = r136984 * r136997;
        double r136999 = r136993 ? r136994 : r136998;
        double r137000 = r136983 ? r136991 : r136999;
        return r137000;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.4
Target34.4
Herbie27.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 57.1

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

    3. Applied flip-+57.1

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

    4. Simplified30.9

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

    if 0.0 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 3.487300811359299e+76

    1. Initial program 4.1

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

    if 3.487300811359299e+76 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))

    1. Initial program 63.7

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

    3. Applied add-exp-log63.7

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

    4. Taylor expanded around 0 44.8

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \le 3.487300811359298841390450553085918164253 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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)))))