Cubic critical

Average Error: 32.6 → 9.8

Time: 4.5s

Precision: binary64

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.019221836402376 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 5.51314038804235395 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{1}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.019221836402376e116

   1. Initial program 52.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around -inf 3.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

   if -1.019221836402376e116 < b < 5.51314038804235395e-45

   1. Initial program 13.3

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

   3. Applied div-inv 13.4

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 13.4

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

   6. Applied times-frac 13.5

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

   7. Simplified 13.5

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

   8. Simplified 13.5

      \[\leadsto \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\frac{1}{a}}\right)\]
   9. Using strategy rm

   10. Applied associate-*l/ 13.5

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

   11. Simplified 13.5

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

   if 5.51314038804235395e-45 < b

   1. Initial program 54.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around inf 7.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.019221836402376 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 5.51314038804235395 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{1}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))