quadm (p42, negative)

Average Error: 34.2 → 6.5

Time: 7.6s

Precision: binary64

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9.560667674887568 \cdot 10^{+134}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.428128095775905 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 2.2264978507093777 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.560667674887568e+134)
   (- (/ c b))
   (if (<= b 4.428128095775905e-281)
     (* 2.0 (/ c (- (sqrt (- (* b b) (* 4.0 (* c a)))) b)))
     (if (<= b 2.2264978507093777e+94)
       (/ (- (- b) (sqrt (+ (* b b) (* c (* a -4.0))))) (* 2.0 a))
       (/ (- b) a)))))

C

double code(double a, double b, double c) {
	return (-b - sqrt((b * b) - (4.0 * (a * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.560667674887568e+134) {
		tmp = -(c / b);
	} else if (b <= 4.428128095775905e-281) {
		tmp = 2.0 * (c / (sqrt((b * b) - (4.0 * (c * a))) - b));
	} else if (b <= 2.2264978507093777e+94) {
		tmp = (-b - sqrt((b * b) + (c * (a * -4.0)))) / (2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.2
Herbie	6.5

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if b < -9.5606676748875676e134

   1. Initial program 62.0

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 1.9

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. Simplified 1.9

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -9.5606676748875676e134 < b < 4.42812809577590528e-281

   1. Initial program 33.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Using strategy rm

   3. Applied flip--_binary64 33.4

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

   4. Simplified 15.8

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

   5. Simplified 15.8

      \[\leadsto \frac{\frac{4 \cdot \left(c \cdot a\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}{2 \cdot a} \]
   6. Using strategy rm

   7. Applied *-un-lft-identity_binary64 15.8

      \[\leadsto \frac{\frac{4 \cdot \left(c \cdot a\right)}{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}}}{2 \cdot a} \]

   8. Applied times-frac_binary64 15.8

      \[\leadsto \frac{\color{blue}{\frac{4}{1} \cdot \frac{c \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}{2 \cdot a} \]

   9. Applied times-frac_binary64 15.8

      \[\leadsto \color{blue}{\frac{\frac{4}{1}}{2} \cdot \frac{\frac{c \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}{a}} \]

   10. Simplified 15.8

       \[\leadsto \color{blue}{2} \cdot \frac{\frac{c \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}{a} \]

   11. Simplified 8.2

       \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\right)} \]

   12. Taylor expanded in b around 0 8.2

       \[\leadsto 2 \cdot \left(1 \cdot \frac{c}{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(c \cdot a\right)}} - b}\right) \]

   13. Simplified 8.2

       \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}} \]

   if 4.42812809577590528e-281 < b < 2.22649785070937769e94

   1. Initial program 9.2

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Using strategy rm

   3. Applied sub-neg_binary64 9.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a} \]

   4. Simplified 9.1

      \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}}}{2 \cdot a} \]

   if 2.22649785070937769e94 < b

   1. Initial program 45.1

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 4.3

      \[\leadsto \frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a} \]

   3. Simplified 4.1

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.560667674887568 \cdot 10^{+134}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.428128095775905 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 2.2264978507093777 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))