The quadratic formula (r2)

Average Error: 34.1 → 6.3

Time: 7.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -7.074176182163289 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.619636124755816 \cdot 10^{-304}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\right)\\ \mathbf{elif}\;b \leq 5.342562827435607 \cdot 10^{+94}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]

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 -7.074176182163289e+117)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 8.619636124755816e-304)
     (* -0.5 (* -4.0 (/ c (- (sqrt (- (* b b) (* 4.0 (* c a)))) b))))
     (if (<= b 5.342562827435607e+94)
       (* -0.5 (/ (+ b (sqrt (- (* b b) (* c (* 4.0 a))))) a))
       (* -0.5 (* 2.0 (- (/ b a) (/ c b))))))))

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 <= -7.074176182163289e+117) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 8.619636124755816e-304) {
		tmp = -0.5 * (-4.0 * (c / (sqrt((b * b) - (4.0 * (c * a))) - b)));
	} else if (b <= 5.342562827435607e+94) {
		tmp = -0.5 * ((b + sqrt((b * b) - (c * (4.0 * a)))) / a);
	} else {
		tmp = -0.5 * (2.0 * ((b / a) - (c / b)));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.1
Target	20.5
Herbie	6.3

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

Derivation

1. Split input into 4 regimes

2. if b < -7.07417618216328877e117

   1. Initial program 61.0

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

   2. Simplified 61.0

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

   3. Taylor expanded around -inf 2.3

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

   if -7.07417618216328877e117 < b < 8.6196361247558162e-304

   1. Initial program 33.4

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

   2. Simplified 33.4

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

   4. Applied flip-+_binary64 33.4

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

   5. Simplified 16.1

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

   7. Applied *-un-lft-identity_binary64 16.1

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

   8. Applied *-un-lft-identity_binary64 16.1

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

   9. Applied times-frac_binary64 16.1

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

   10. Applied times-frac_binary64 16.1

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

   11. Simplified 16.1

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

   12. Simplified 7.6

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

   14. Applied frac-2neg_binary64 7.6

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

   15. Simplified 7.6

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

   if 8.6196361247558162e-304 < b < 5.34256282743560728e94

   1. Initial program 8.9

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

   2. Simplified 8.9

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

   4. Applied associate-*r*_binary64 8.9

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

   if 5.34256282743560728e94 < b

   1. Initial program 45.4

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

   2. Simplified 45.4

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

   3. Taylor expanded around inf 4.2

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

   4. Simplified 4.2

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

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.074176182163289 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.619636124755816 \cdot 10^{-304}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\right)\\ \mathbf{elif}\;b \leq 5.342562827435607 \cdot 10^{+94}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021113 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))