math.sqrt on complex, real part

Average Error: 38.9 → 10.4

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;t_0 \leq -6.5239109072974635 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{re \cdot re}{re - \mathsf{hypot}\left(re, im\right)} - \frac{\mathsf{hypot}\left(re, im\right)}{\frac{re}{\mathsf{hypot}\left(re, im\right)} + -1}\right)}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

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 -6.5239109072974635e-307)
     (*
      0.5
      (sqrt
       (*
        2.0
        (-
         (/ (* re re) (- re (hypot re im)))
         (/ (hypot re im) (+ (/ re (hypot re im)) -1.0))))))
     (if (<= t_0 0.0)
       (* 0.5 (sqrt (* 2.0 (* (/ (* im im) re) -0.5))))
       (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))))

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 <= -6.5239109072974635e-307) {
		tmp = 0.5 * sqrt(2.0 * (((re * re) / (re - hypot(re, im))) - (hypot(re, im) / ((re / hypot(re, im)) + -1.0))));
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt(2.0 * (((im * im) / re) * -0.5));
	} else {
		tmp = 0.5 * sqrt(2.0 * (re + hypot(re, im)));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.9
Target	33.9
Herbie	10.4

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -6.5239109072974635e-307

   1. Initial program 64.0

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

   2. Simplified 32.5

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

   4. Applied flip-+_binary64 61.1

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

   5. Applied div-sub_binary64 61.1

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

   6. Simplified 33.4

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

   if -6.5239109072974635e-307 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 57.8

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

   2. Simplified 57.6

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

   3. Taylor expanded around -inf 31.5

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

   4. Simplified 31.5

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

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

   1. Initial program 35.3

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

   2. Simplified 6.5

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

   4. Applied *-un-lft-identity_binary64 6.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{1 \cdot \mathsf{hypot}\left(re, im\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -6.5239109072974635 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{re \cdot re}{re - \mathsf{hypot}\left(re, im\right)} - \frac{\mathsf{hypot}\left(re, im\right)}{\frac{re}{\mathsf{hypot}\left(re, im\right)} + -1}\right)}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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)))))