math.sqrt on complex, real part

Average Error: 38.7 → 12.2

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.99259449188669329 \cdot 10^{225} \lor \neg \left(re \le -1.1569629420680933 \cdot 10^{160} \lor \neg \left(re \le -1.7053538714353348 \cdot 10^{-44}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]

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 temp;
	if (((re <= -1.9925944918866933e+225) || !((re <= -1.1569629420680933e+160) || !(re <= -1.7053538714353348e-44)))) {
		temp = (0.5 * sqrt((2.0 * ((0.0 + pow(im, 2.0)) / (hypot(re, im) - re)))));
	} else {
		temp = (0.5 * sqrt((2.0 * (1.0 * (re + hypot(re, im))))));
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.7
Herbie	12.2

\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if re < -1.9925944918866933e+225 or -1.1569629420680933e+160 < re < -1.7053538714353348e-44

   1. Initial program 53.1

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

   3. Applied flip-+ 53.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Simplified 36.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

   5. Simplified 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]

   if -1.9925944918866933e+225 < re < -1.1569629420680933e+160 or -1.7053538714353348e-44 < re

   1. Initial program 34.2

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

   3. Applied *-un-lft-identity 34.2

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

   4. Applied *-un-lft-identity 34.2

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

   5. Applied distribute-lft-out 34.2

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

   6. Simplified 6.2

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

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.99259449188669329 \cdot 10^{225} \lor \neg \left(re \le -1.1569629420680933 \cdot 10^{160} \lor \neg \left(re \le -1.7053538714353348 \cdot 10^{-44}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))