Quadratic roots, full range

Average Error: 34.1 → 6.6

Time: 8.2min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.6269445204268617 \cdot 10^{132}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.00283392729217193 \cdot 10^{-183}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{elif}\;b \le 3.9872843243560293 \cdot 10^{131}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -4.6269445204268617e132

   1. Initial program 56.7

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

   2. Simplified 56.7

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

   3. Taylor expanded around -inf 3.0

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

   4. Simplified 3.0

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

   if -4.6269445204268617e132 < b < -1.00283392729217193e-183

   1. Initial program 6.8

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

   2. Simplified 6.8

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

   4. Applied *-un-lft-identity 6.8

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

   5. Applied div-inv 6.8

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

   6. Applied times-frac 7.0

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

   7. Simplified 7.0

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

   if -1.00283392729217193e-183 < b < 3.9872843243560293e131

   1. Initial program 31.2

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

   2. Simplified 31.2

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

   4. Applied *-un-lft-identity 31.2

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

   5. Applied div-inv 31.2

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

   6. Applied times-frac 31.2

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

   7. Simplified 31.2

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

   9. Applied flip-- 31.4

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

   10. Applied associate-*l/ 31.4

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

   11. Simplified 15.2

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

   12. Taylor expanded around 0 9.9

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

   if 3.9872843243560293e131 < b

   1. Initial program 61.9

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

   2. Simplified 61.9

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

   3. Taylor expanded around inf 1.6

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.6269445204268617 \cdot 10^{132}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.00283392729217193 \cdot 10^{-183}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{elif}\;b \le 3.9872843243560293 \cdot 10^{131}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))