math.sqrt on complex, real part

Average Error: 38.2 → 9.6

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;t_0 \leq -7.31236209128368 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{0.5}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (sqrt (+ (* re re) (* im im))))))
   (if (<= t_0 -7.31236209128368e-302)
     (*
      0.5
      (sqrt (* 2.0 (fma (* (cbrt re) (cbrt re)) (cbrt re) (hypot re im)))))
     (if (<= t_0 0.0)
       (* 0.5 (sqrt (* 2.0 (* -0.5 (/ (pow im 2.0) re)))))
       (* 0.5 (pow (* 2.0 (+ re (hypot re im))) 0.5))))))

C

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 t_0 = re + sqrt((re * re) + (im * im));
	double tmp;
	if (t_0 <= -7.31236209128368e-302) {
		tmp = 0.5 * sqrt(2.0 * fma((cbrt(re) * cbrt(re)), cbrt(re), hypot(re, im)));
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt(2.0 * (-0.5 * (pow(im, 2.0) / re)));
	} else {
		tmp = 0.5 * pow((2.0 * (re + hypot(re, im))), 0.5);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.2
Target	33.4
Herbie	9.6

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -7.3123620912836802e-302

   1. Initial program 64.0

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

   2. Simplified 28.4

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

   3. Applied add-cube-cbrt_binary64 28.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}} + \mathsf{hypot}\left(re, im\right)\right)} \]

   4. Applied fma-def_binary64 28.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   if -7.3123620912836802e-302 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 57.7

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

   2. Simplified 57.2

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

   3. Taylor expanded in re around -inf 31.2

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

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

   1. Initial program 34.7

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

   2. Simplified 6.1

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

   3. Applied sqrt-prod_binary64 6.4

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

   4. Applied pow1/2_binary64 6.4

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

   5. Applied pow1/2_binary64 6.4

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

   6. Applied pow-prod-down_binary64 6.1

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -7.31236209128368 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{0.5}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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)))))