The quadratic formula (r1)

Average Error: 34.6 → 6.2

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.7129290315884307 \cdot 10^{136}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.93459722336773847 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.8799435483517381 \cdot 10^{126}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.1
Herbie	6.2

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

Derivation

1. Split input into 4 regimes

2. if b < -4.7129290315884307e136

   1. Initial program 58.2

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

   2. Taylor expanded around -inf 2.3

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

   3. Simplified 2.3

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

   if -4.7129290315884307e136 < b < -3.93459722336773847e-278

   1. Initial program 8.9

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

   if -3.93459722336773847e-278 < b < 2.8799435483517381e126

   1. Initial program 32.1

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

   3. Applied flip-+ 32.2

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

   4. Simplified 15.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
   5. Using strategy rm

   6. Applied clear-num 15.2

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

   7. Simplified 14.2

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

   8. Taylor expanded around 0 8.4

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

   10. Applied associate-/r* 8.1

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

   11. Simplified 8.0

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

   if 2.8799435483517381e126 < b

   1. Initial program 61.6

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

   2. Taylor expanded around inf 1.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.7129290315884307 \cdot 10^{136}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.93459722336773847 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.8799435483517381 \cdot 10^{126}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020169 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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