The quadratic formula (r1)

Average Error: 34.3 → 17.5

Time: 8.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1675890391810646 \cdot 10^{+147}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.346684968652724 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.857075353436113 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b}\\ \end{array}\]

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 -4.1675890391810646e+147)
   (/ (- b) (* a 2.0))
   (if (<= b -1.346684968652724e-176)
     (/
      (*
       (sqrt (- (sqrt (- (* b b) (* (* a 4.0) c))) b))
       (sqrt (- (sqrt (- (* b b) (* (* a 4.0) c))) b)))
      (* a 2.0))
     (if (<= b 2.857075353436113e+153)
       (* -2.0 (/ c (+ b (sqrt (- (* b b) (* (* a 4.0) c))))))
       (* -2.0 (/ 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 <= -4.1675890391810646e+147) {
		tmp = -b / (a * 2.0);
	} else if (b <= -1.346684968652724e-176) {
		tmp = (sqrt(sqrt((b * b) - ((a * 4.0) * c)) - b) * sqrt(sqrt((b * b) - ((a * 4.0) * c)) - b)) / (a * 2.0);
	} else if (b <= 2.857075353436113e+153) {
		tmp = -2.0 * (c / (b + sqrt((b * b) - ((a * 4.0) * c))));
	} else {
		tmp = -2.0 * (c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.3
Target	20.6
Herbie	17.5

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

Derivation

1. Split input into 4 regimes

2. if b < -4.1675890391810646e147

   1. Initial program 61.7

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

   2. Simplified 61.7

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

   3. Taylor expanded around 0 52.3

      \[\leadsto \frac{\color{blue}{0} - b}{a \cdot 2}\]

   if -4.1675890391810646e147 < b < -1.346684968652724e-176

   1. Initial program 7.1

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

   2. Simplified 7.1

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

   4. Applied add-sqr-sqrt_binary64_1692 7.4

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

   if -1.346684968652724e-176 < b < 2.8570753534361131e153

   1. Initial program 30.9

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

   2. Simplified 30.9

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

   4. Applied flip--_binary64_1733 31.0

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

   5. Simplified 15.1

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

   6. Simplified 15.1

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

   8. Applied *-un-lft-identity_binary64_1707 15.1

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

   9. Applied times-frac_binary64_1702 13.3

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

   10. Applied times-frac_binary64_1702 8.6

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

   11. Simplified 8.6

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

   12. Simplified 8.6

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

   if 2.8570753534361131e153 < b

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   4. Applied flip--_binary64_1733 63.9

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

   5. Simplified 37.9

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

   6. Simplified 37.9

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

   8. Applied *-un-lft-identity_binary64_1707 37.9

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

   9. Applied times-frac_binary64_1702 37.7

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

   10. Applied times-frac_binary64_1702 37.7

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

   11. Simplified 37.7

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

   12. Simplified 37.7

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

   13. Taylor expanded around 0 32.5

       \[\leadsto 1 \cdot \left(-2 \cdot \frac{c}{b + \color{blue}{0}}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1675890391810646 \cdot 10^{+147}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.346684968652724 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.857075353436113 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020270 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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

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