quad2m (problem 3.2.1, negative)

Average Error: 34.3 → 6.7

Time: 10.7s

Precision: binary64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.0591588073931438 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6106223235737979 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 2.841423602214126 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.0591588073931438e+154)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.6106223235737979e-307)
     (/ c (- (sqrt (- (* b_2 b_2) (* c a))) b_2))
     (if (<= b_2 2.841423602214126e+70)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (/ (* b_2 -2.0) a)))))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt((b_2 * b_2) - (a * c))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.0591588073931438e+154) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.6106223235737979e-307) {
		tmp = c / (sqrt((b_2 * b_2) - (c * a)) - b_2);
	} else if (b_2 <= 2.841423602214126e+70) {
		tmp = (-b_2 - sqrt((b_2 * b_2) - (c * a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.05915880739314384e154

   1. Initial program 64.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip--_binary64 64.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 36.9

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 36.9

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity_binary64 36.9

      \[\leadsto \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

   8. Applied *-un-lft-identity_binary64 36.9

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{1 \cdot a}\]

   9. Applied times-frac_binary64 36.9

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}}\]

   10. Simplified 36.9

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

   11. Simplified 36.8

       \[\leadsto 1 \cdot \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

   12. Taylor expanded around -inf 0.8

       \[\leadsto 1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b_2}\right)}\]

   if -1.05915880739314384e154 < b_2 < 1.6106223235737979e-307

   1. Initial program 35.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip--_binary64 35.3

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 16.2

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 16.2

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity_binary64 16.2

      \[\leadsto \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

   8. Applied *-un-lft-identity_binary64 16.2

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{1 \cdot a}\]

   9. Applied times-frac_binary64 16.2

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}}\]

   10. Simplified 16.2

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

   11. Simplified 8.4

       \[\leadsto 1 \cdot \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

   if 1.6106223235737979e-307 < b_2 < 2.84142360221412621e70

   1. Initial program 9.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   if 2.84142360221412621e70 < b_2

   1. Initial program 40.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 4.7

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.0591588073931438 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6106223235737979 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 2.841423602214126 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))