Average Error: 33.8 → 15.5
Time: 1.0m
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.1606837337009755 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 3}\\ \mathbf{elif}\;b \le 2.971744846090254 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 3\right) \cdot \left(-c\right)}{3 \cdot \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < 1.1606837337009755e-150

    1. Initial program 20.7

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

    2. Initial simplification20.7

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

    4. Applied *-un-lft-identity20.7

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

    5. Applied associate-/l*20.7

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

    7. Applied associate-/r/20.8

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

    9. Applied associate-*l/20.7

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

    10. Simplified20.7

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

    if 1.1606837337009755e-150 < b < 2.971744846090254e+109

    1. Initial program 39.3

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

    2. Initial simplification39.3

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

    4. Applied associate-/r*39.3

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

    6. Applied flip--39.4

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

    7. Applied associate-/l/39.4

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

    8. Simplified15.5

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

    if 2.971744846090254e+109 < b

    1. Initial program 59.5

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

    2. Initial simplification59.6

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

    4. Applied *-un-lft-identity59.6

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

    5. Applied associate-/l*59.6

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

    6. Taylor expanded around 0 2.9

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.1606837337009755 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 3}\\ \mathbf{elif}\;b \le 2.971744846090254 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 3\right) \cdot \left(-c\right)}{3 \cdot \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c}}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))