quad2m (problem 3.2.1, negative)

Average Error: 34.7 → 6.7

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.14914207805587318 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.3795045532527556 \cdot 10^{-190}:\\ \;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.94983884541345716 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

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 < -3.149142078055873e+153

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 1.4

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

   if -3.149142078055873e+153 < b_2 < 5.379504553252756e-190

   1. Initial program 31.6

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

   3. Applied flip-- 31.7

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

   4. Simplified 16.3

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

   5. Simplified 16.3

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

   7. Applied *-un-lft-identity 16.3

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

   8. Applied associate-/r* 16.3

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

   9. Simplified 14.6

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

   11. Applied clear-num 14.6

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

   12. Simplified 9.8

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}\]
   13. Using strategy rm

   14. Applied *-un-lft-identity 9.8

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

   15. Applied *-un-lft-identity 9.8

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

   16. Applied times-frac 9.8

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

   17. Applied add-cube-cbrt 9.8

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\]

   18. Applied times-frac 9.8

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}\]

   19. Simplified 9.8

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\]

   20. Simplified 9.4

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

   if 5.379504553252756e-190 < b_2 < 5.949838845413457e+107

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt 7.5

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   if 5.949838845413457e+107 < b_2

   1. Initial program 50.1

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

   2. Taylor expanded around inf 3.8

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.14914207805587318 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.3795045532527556 \cdot 10^{-190}:\\ \;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.94983884541345716 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))