math.sqrt on complex, real part

Average Error: 38.8 → 10.1

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ t_1 := re + \mathsf{hypot}\left(re, im\right)\\ \mathbf{if}\;t_0 \leq -2.908944228556174 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot t_1\right)}^{0.5}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{t_1}\right)\\ \end{array} \]

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))))) (t_1 (+ re (hypot re im))))
   (if (<= t_0 -2.908944228556174e-287)
     (* 0.5 (pow (* 2.0 t_1) 0.5))
     (if (<= t_0 0.0)
       (* 0.5 (sqrt (* 2.0 (* -0.5 (/ (pow im 2.0) re)))))
       (* 0.5 (* (sqrt 2.0) (sqrt t_1)))))))

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 t_1 = re + hypot(re, im);
	double tmp;
	if (t_0 <= -2.908944228556174e-287) {
		tmp = 0.5 * pow((2.0 * t_1), 0.5);
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt(2.0 * (-0.5 * (pow(im, 2.0) / re)));
	} else {
		tmp = 0.5 * (sqrt(2.0) * sqrt(t_1));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.8
Target	33.6
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -2.908944228556174e-287

   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. Applied add-sqr-sqrt_binary64 32.8

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

   4. Applied sqrt-prod_binary64 32.9

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

   5. Simplified 32.6

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

   6. Applied pow1_binary64 32.6

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

   7. Applied sqrt-pow1_binary64 32.6

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

   8. Applied pow1_binary64 32.6

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

   9. Applied sqrt-pow1_binary64 32.6

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

   10. Applied pow-prod-down_binary64 32.5

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

   if -2.908944228556174e-287 < (+.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 55.9

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

   3. Taylor expanded in re around -inf 29.4

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

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

   1. Initial program 35.3

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

   2. Simplified 6.3

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

   3. Applied add-sqr-sqrt_binary64 7.5

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

   4. Applied sqrt-prod_binary64 7.6

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

   5. Simplified 6.7

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -2.908944228556174 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + \mathsf{hypot}\left(re, im\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))))