math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.9 → 6.3

Time: 3.8s

Precision: binary64

\[im > 0\]

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (let* ((t_0 (sqrt (sqrt 2.0))))
     (* 0.5 (/ (* (sqrt 0.5) (* t_0 (* im t_0))) (sqrt re))))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

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 tmp;
	if ((sqrt((re * re) + (im * im)) - re) <= 0.0) {
		double t_0_1 = sqrt(sqrt(2.0));
		tmp = 0.5 * ((sqrt(0.5) * (t_0_1 * (im * t_0_1))) / sqrt(re));
	} else {
		tmp = 0.5 * sqrt(2.0 * (hypot(re, im) - re));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.0

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

   2. Simplified 51.6

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

   3. Taylor expanded in im around 0 7.0

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

   4. Applied add-sqr-sqrt_binary64 6.5

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

   5. Applied associate-*l*_binary64 6.6

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

   6. Applied sqrt-div_binary64 6.6

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

   7. Applied associate-*r/_binary64 6.5

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

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

   1. Initial program 35.1

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

   2. Simplified 6.3

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2021313 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))