quad2m (problem 3.2.1, negative)

Average Error: 34.2 → 6.5

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.3488319803265095 \cdot 10^{156}:\\ \;\;\;\;\frac{c}{\left(a \cdot \frac{\frac{1}{2}}{\frac{b_2}{c}} - b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \le 1.31643909605529108 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 4.63496539510710479 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \end{array}\]

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 < -5.3488319803265095e156

   1. Initial program 63.9

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

   3. Applied flip-- 62.9

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

   4. Simplified 38.7

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

   5. Simplified 38.7

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

   7. Applied *-un-lft-identity 38.7

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

   8. Applied *-un-lft-identity 38.7

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

   9. Applied times-frac 38.7

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

   10. Simplified 38.7

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

   11. Simplified 38.7

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

   12. Taylor expanded around -inf 6.3

       \[\leadsto 1 \cdot \frac{c}{\color{blue}{\left(\frac{1}{2} \cdot \frac{a \cdot c}{b_2} - b_2\right)} - b_2}\]

   13. Simplified 1.2

       \[\leadsto 1 \cdot \frac{c}{\color{blue}{\left(a \cdot \frac{\frac{1}{2}}{\frac{b_2}{c}} - b_2\right)} - b_2}\]

   if -5.3488319803265095e156 < b_2 < 1.31643909605529108e-307

   1. Initial program 34.2

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

   3. Applied flip-- 34.1

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

   4. Simplified 15.2

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

   5. Simplified 15.2

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

   7. Applied *-un-lft-identity 15.2

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

   8. Applied *-un-lft-identity 15.2

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

   9. Applied times-frac 15.2

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

   10. Simplified 15.2

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

   11. Simplified 8.1

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

   if 1.31643909605529108e-307 < b_2 < 4.63496539510710479e117

   1. Initial program 9.0

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

   3. Applied clear-num 9.2

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

   if 4.63496539510710479e117 < b_2

   1. Initial program 51.4

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

   2. Taylor expanded around inf 3.1

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

   3. Simplified 3.1

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.3488319803265095 \cdot 10^{156}:\\ \;\;\;\;\frac{c}{\left(a \cdot \frac{\frac{1}{2}}{\frac{b_2}{c}} - b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \le 1.31643909605529108 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 4.63496539510710479 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (neg b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))