Average Error: 38.7 → 10.9
Time: 4.8s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;\begin{array}{l} t_0 := {\left(\frac{-1}{re}\right)}^{0.5} \cdot e^{\log im}\\ 0.5 \cdot \mathsf{fma}\left(\frac{im \cdot im}{re \cdot re} \cdot t_0, -0.125, t_0\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)\\ \end{array} \]
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(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\
\;\;\;\;\begin{array}{l}
t_0 := {\left(\frac{-1}{re}\right)}^{0.5} \cdot e^{\log im}\\
0.5 \cdot \mathsf{fma}\left(\frac{im \cdot im}{re \cdot re} \cdot t_0, -0.125, t_0\right)
\end{array}\\

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


\end{array}

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (let* ((t_0 (* (pow (/ -1.0 re) 0.5) (exp (log im)))))
     (* 0.5 (fma (* (/ (* im im) (* re re)) t_0) -0.125 t_0)))
   (* 0.5 (* (sqrt (sqrt 2.0)) (sqrt (* (sqrt 2.0) (+ re (hypot re im))))))))
double code(double re, double im) {
	return 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) + re));
}
↓
double code(double re, double im) {
	double tmp;
	if (sqrt(2.0 * (re + sqrt((re * re) + (im * im)))) <= 0.0) {
		double t_0_1 = pow((-1.0 / re), 0.5) * exp(log(im));
		tmp = 0.5 * fma((((im * im) / (re * re)) * t_0_1), -0.125, t_0_1);
	} else {
		tmp = 0.5 * (sqrt(sqrt(2.0)) * sqrt(sqrt(2.0) * (re + hypot(re, im))));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.7
Target33.7
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 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. Simplified57.2

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

    3. Applied sqrt-prod_binary6457.2

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

    4. Applied add-sqr-sqrt_binary6457.2

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

    5. Applied associate-*l*_binary6457.2

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

    6. Applied pow1_binary6457.2

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

    7. Applied sqrt-pow1_binary6457.2

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

    8. Applied pow1_binary6457.2

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

    9. Applied sqrt-pow1_binary6457.2

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

    10. Applied pow-prod-down_binary6457.2

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

    11. Applied pow1_binary6457.2

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

    12. Applied sqrt-pow1_binary6457.2

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

    13. Applied pow-prod-down_binary6457.2

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

    14. Simplified57.2

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

    15. Taylor expanded in re around -inf 28.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{0.5 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)} - 0.125 \cdot \frac{e^{0.5 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)} \cdot {im}^{2}}{{re}^{2}}\right)} \]

    16. Simplified34.8

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{im \cdot im}{re \cdot re} \cdot \left({\left(\frac{-1}{re}\right)}^{0.5} \cdot e^{\log im}\right), -0.125, {\left(\frac{-1}{re}\right)}^{0.5} \cdot e^{\log im}\right)} \]

    if 0.0 < (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

    1. Initial program 36.1

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

    2. Simplified7.4

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

    3. Applied sqrt-prod_binary647.7

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

    4. Applied add-sqr-sqrt_binary647.8

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

    5. Applied associate-*l*_binary647.7

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

    6. Applied sqrt-unprod_binary647.6

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{im \cdot im}{re \cdot re} \cdot \left({\left(\frac{-1}{re}\right)}^{0.5} \cdot e^{\log im}\right), -0.125, {\left(\frac{-1}{re}\right)}^{0.5} \cdot e^{\log im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))