Average Error: 31.7 → 7.3
Time: 2.3s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 1.542344810860402 \cdot 10^{-147}:\\ \;\;\;\;\frac{im}{\sqrt[3]{re} \cdot \left(\sqrt[3]{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right)} \cdot \left(-0.5 \cdot \frac{im}{\sqrt[3]{re}}\right) - re\\ \mathbf{elif}\;im \leq 2.040124260634893 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 1.542344810860402 \cdot 10^{-147}:\\
\;\;\;\;\frac{im}{\sqrt[3]{re} \cdot \left(\sqrt[3]{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right)} \cdot \left(-0.5 \cdot \frac{im}{\sqrt[3]{re}}\right) - re\\

\mathbf{elif}\;im \leq 2.040124260634893 \cdot 10^{+114}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;im\\

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im 1.542344810860402e-147)
   (-
    (*
     (/ im (* (cbrt re) (* (cbrt (* (cbrt re) (cbrt re))) (cbrt (cbrt re)))))
     (* -0.5 (/ im (cbrt re))))
    re)
   (if (<= im 2.040124260634893e+114) (sqrt (+ (* re re) (* im im))) im)))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (im <= 1.542344810860402e-147) {
		tmp = ((im / (cbrt(re) * (cbrt(cbrt(re) * cbrt(re)) * cbrt(cbrt(re))))) * (-0.5 * (im / cbrt(re)))) - re;
	} else if (im <= 2.040124260634893e+114) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = im;
	}
	return tmp;
}

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 im < 1.5423448108604021e-147

    1. Initial program 32.5

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

      \[\leadsto \color{blue}{\frac{im \cdot im}{re} \cdot -0.5 - re}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary648.1

      \[\leadsto \frac{im \cdot im}{\color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}} \cdot -0.5 - re\]

    6. Applied times-frac_binary645.4

      \[\leadsto \color{blue}{\left(\frac{im}{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \frac{im}{\sqrt[3]{re}}\right)} \cdot -0.5 - re\]

    7. Applied associate-*l*_binary645.4

      \[\leadsto \color{blue}{\frac{im}{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \left(\frac{im}{\sqrt[3]{re}} \cdot -0.5\right)} - re\]

    8. Simplified5.4

      \[\leadsto \frac{im}{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \color{blue}{\left(-0.5 \cdot \frac{im}{\sqrt[3]{re}}\right)} - re\]
    9. Using strategy rm

    10. Applied add-cube-cbrt_binary645.4

      \[\leadsto \frac{im}{\sqrt[3]{re} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}}} \cdot \left(-0.5 \cdot \frac{im}{\sqrt[3]{re}}\right) - re\]

    11. Applied cbrt-prod_binary645.4

      \[\leadsto \frac{im}{\sqrt[3]{re} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right)}} \cdot \left(-0.5 \cdot \frac{im}{\sqrt[3]{re}}\right) - re\]

    if 1.5423448108604021e-147 < im < 2.0401242606348929e114

    1. Initial program 11.6

      \[\sqrt{re \cdot re + im \cdot im}\]

    if 2.0401242606348929e114 < im

    1. Initial program 53.1

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around 0 5.4

      \[\leadsto \color{blue}{im}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.542344810860402 \cdot 10^{-147}:\\ \;\;\;\;\frac{im}{\sqrt[3]{re} \cdot \left(\sqrt[3]{\sqrt[3]{re} \cdot \sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right)} \cdot \left(-0.5 \cdot \frac{im}{\sqrt[3]{re}}\right) - re\\ \mathbf{elif}\;im \leq 2.040124260634893 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))