quadm (p42, negative)

Average Error: 33.9 → 6.6

Time: 5.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.063829179971075 \cdot 10^{+73}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -6.510183985272386 \cdot 10^{-301}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \leq 7.922796939806533 \cdot 10^{+99}:\\ \;\;\;\;-0.5 \cdot \left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\ \end{array}\]

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 -6.063829179971075e+73)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b -6.510183985272386e-301)
     (* -0.5 (/ (* c 4.0) (- b (sqrt (- (* b b) (* 4.0 (* c a)))))))
     (if (<= b 7.922796939806533e+99)
       (* -0.5 (* (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) (/ 1.0 a)))
       (* -0.5 (* 2.0 (/ 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 <= -6.063829179971075e+73) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= -6.510183985272386e-301) {
		tmp = -0.5 * ((c * 4.0) / (b - sqrt((b * b) - (4.0 * (c * a)))));
	} else if (b <= 7.922796939806533e+99) {
		tmp = -0.5 * ((b + sqrt((b * b) - (4.0 * (c * a)))) * (1.0 / a));
	} else {
		tmp = -0.5 * (2.0 * (b / a));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.9
Target	20.5
Herbie	6.6

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

Derivation

1. Split input into 4 regimes

2. if b < -6.06382917997107451e73

   1. Initial program 57.6

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

   2. Simplified 57.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

   3. Taylor expanded around -inf 3.2

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)}\]

   if -6.06382917997107451e73 < b < -6.5101839852723858e-301

   1. Initial program 31.2

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

   2. Simplified 31.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]
   3. Using strategy rm

   4. Applied div-inv_binary64_764 31.2

      \[\leadsto -0.5 \cdot \color{blue}{\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}\right)}\]
   5. Using strategy rm

   6. Applied flip-+_binary64_741 31.2

      \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{1}{a}\right)\]

   7. Applied associate-*l/_binary64_710 31.2

      \[\leadsto -0.5 \cdot \color{blue}{\frac{\left(b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   8. Simplified 15.2

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_767 15.2

       \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   11. Applied times-frac_binary64_773 15.2

       \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   12. Simplified 15.2

       \[\leadsto -0.5 \cdot \frac{\color{blue}{4} \cdot \frac{a \cdot c}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   13. Simplified 8.2

       \[\leadsto -0.5 \cdot \frac{4 \cdot \color{blue}{c}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   if -6.5101839852723858e-301 < b < 7.92279693980653335e99

   1. Initial program 9.3

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

   2. Simplified 9.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]
   3. Using strategy rm

   4. Applied div-inv_binary64_764 9.5

      \[\leadsto -0.5 \cdot \color{blue}{\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}\right)}\]

   if 7.92279693980653335e99 < b

   1. Initial program 47.6

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

   2. Simplified 47.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]
   3. Using strategy rm

   4. Applied div-inv_binary64_764 47.7

      \[\leadsto -0.5 \cdot \color{blue}{\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}\right)}\]
   5. Using strategy rm

   6. Applied flip-+_binary64_741 63.5

      \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{1}{a}\right)\]

   7. Applied associate-*l/_binary64_710 63.5

      \[\leadsto -0.5 \cdot \color{blue}{\frac{\left(b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   8. Simplified 62.6

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_767 62.6

       \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   11. Applied times-frac_binary64_773 62.6

       \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   12. Simplified 62.6

       \[\leadsto -0.5 \cdot \frac{\color{blue}{4} \cdot \frac{a \cdot c}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   13. Simplified 62.4

       \[\leadsto -0.5 \cdot \frac{4 \cdot \color{blue}{c}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   14. Taylor expanded around 0 3.6

       \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.063829179971075 \cdot 10^{+73}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -6.510183985272386 \cdot 10^{-301}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \leq 7.922796939806533 \cdot 10^{+99}:\\ \;\;\;\;-0.5 \cdot \left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected #f

  :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)))