quad2m (problem 3.2.1, negative)

Average Error: 34.0 → 7.0

Time: 5.9s

Precision: 64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.5927674402090787 \cdot 10^{66}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.5139323844298838 \cdot 10^{-270}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 1.0676041056623592 \cdot 10^{96}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.592767440209079e+66

   1. Initial program 57.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 3.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   if -9.592767440209079e+66 < b_2 < 2.513932384429884e-270

   1. Initial program 29.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-- 29.3

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 16.2

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 16.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

   8. Applied associate-/r* 16.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}{a}}\]

   9. Simplified 14.2

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
   10. Using strategy rm

   11. Applied clear-num 14.1

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}}\]

   12. Simplified 10.1

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}\]

   if 2.513932384429884e-270 < b_2 < 1.0676041056623592e+96

   1. Initial program 8.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   if 1.0676041056623592e+96 < b_2

   1. Initial program 46.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.5927674402090787 \cdot 10^{66}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.5139323844298838 \cdot 10^{-270}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 1.0676041056623592 \cdot 10^{96}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))