Average Error: 37.5 → 13.3
Time: 18.0s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r6689944 = 0.5;
        double r6689945 = 2.0;
        double r6689946 = re;
        double r6689947 = r6689946 * r6689946;
        double r6689948 = im;
        double r6689949 = r6689948 * r6689948;
        double r6689950 = r6689947 + r6689949;
        double r6689951 = sqrt(r6689950);
        double r6689952 = r6689951 + r6689946;
        double r6689953 = r6689945 * r6689952;
        double r6689954 = sqrt(r6689953);
        double r6689955 = r6689944 * r6689954;
        return r6689955;
}
↓
double f(double re, double im) {
        double r6689956 = re;
        double r6689957 = im;
        double r6689958 = hypot(r6689956, r6689957);
        double r6689959 = r6689956 + r6689958;
        double r6689960 = 2.0;
        double r6689961 = r6689959 * r6689960;
        double r6689962 = sqrt(r6689961);
        double r6689963 = 0.5;
        double r6689964 = r6689962 * r6689963;
        return r6689964;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target32.8
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.5

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) + re\right) \cdot 2.0}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))