The quadratic formula (r2)

Average Error: 34.3 → 6.8

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9002502210936553 \cdot 10^{+140}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 4.860998560804722 \cdot 10^{-178}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\\ \mathbf{elif}\;b \leq 6.414833710235507 \cdot 10^{+29}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]

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 -1.9002502210936553e+140)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 4.860998560804722e-178)
     (* -0.5 (* 4.0 (/ c (- b (sqrt (+ (* b b) (* a (* c -4.0))))))))
     (if (<= b 6.414833710235507e+29)
       (* -0.5 (/ 1.0 (/ a (+ b (sqrt (- (* b b) (* 4.0 (* c 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 <= -1.9002502210936553e+140) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 4.860998560804722e-178) {
		tmp = -0.5 * (4.0 * (c / (b - sqrt((b * b) + (a * (c * -4.0))))));
	} else if (b <= 6.414833710235507e+29) {
		tmp = -0.5 * (1.0 / (a / (b + sqrt((b * b) - (4.0 * (c * 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	34.3
Target	21.1
Herbie	6.8

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

Derivation

1. Split input into 4 regimes

2. if b < -1.9002502210936553e140

   1. Initial program 62.6

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

   2. Simplified 62.6

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

   3. Taylor expanded around -inf 2.1

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

   if -1.9002502210936553e140 < b < 4.86099856080472191e-178

   1. Initial program 30.3

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

   2. Simplified 30.2

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

   4. Applied flip-+_binary64_1085 30.4

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

   5. Simplified 15.6

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

   7. Applied *-un-lft-identity_binary64_1111 15.6

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

   8. Applied *-un-lft-identity_binary64_1111 15.6

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

   9. Applied times-frac_binary64_1117 15.6

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

   10. Applied times-frac_binary64_1117 15.6

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

   11. Simplified 15.6

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

   12. Simplified 9.3

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

   14. Applied sub-neg_binary64_1104 9.3

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

   15. Simplified 9.3

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

   if 4.86099856080472191e-178 < b < 6.41483371023550709e29

   1. Initial program 6.8

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

   2. Simplified 6.8

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

   4. Applied clear-num_binary64_1110 7.0

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

   if 6.41483371023550709e29 < b

   1. Initial program 36.4

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

   2. Simplified 36.4

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

   3. Taylor expanded around inf 6.1

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

   4. Simplified 6.1

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9002502210936553 \cdot 10^{+140}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 4.860998560804722 \cdot 10^{-178}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\\ \mathbf{elif}\;b \leq 6.414833710235507 \cdot 10^{+29}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(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)))