The quadratic formula (r2)

Average Error: 34.1 → 8.5

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.1394562579328032 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.93284044029756146 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 7.1657755535705989 \cdot 10^{126}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.1
Target	20.7
Herbie	8.5

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -4.1394562579328032e101

   1. Initial program 59.5

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

   2. Taylor expanded around -inf 2.0

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

   if -4.1394562579328032e101 < b < -1.93284044029756146e-100

   1. Initial program 42.9

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

   3. Applied flip-- 42.9

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

   4. Simplified 14.5

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

   5. Simplified 14.5

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

   7. Applied *-un-lft-identity 14.5

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

   8. Applied *-un-lft-identity 14.5

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

   9. Applied times-frac 14.5

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

   10. Applied times-frac 14.5

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

   11. Simplified 14.5

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

   12. Simplified 12.9

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

   if -1.93284044029756146e-100 < b < 7.1657755535705989e126

   1. Initial program 11.7

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

   if 7.1657755535705989e126 < b

   1. Initial program 54.0

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

   3. Applied flip-- 63.7

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

   4. Simplified 63.7

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

   5. Simplified 63.7

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

   7. Applied *-un-lft-identity 63.7

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

   8. Applied *-un-lft-identity 63.7

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

   9. Applied times-frac 63.7

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

   10. Applied times-frac 63.7

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

   11. Simplified 63.7

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

   12. Simplified 62.7

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

   14. Applied *-un-lft-identity 62.7

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

   15. Applied times-frac 62.7

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

   16. Simplified 62.7

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

   17. Simplified 62.5

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

   18. Taylor expanded around 0 3.2

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.1394562579328032 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.93284044029756146 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 7.1657755535705989 \cdot 10^{126}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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