Average Error: 33.5 → 11.9
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.378869344776828 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{3} \cdot \frac{b}{a}\\ \mathbf{if}\;b \le 2.732419256347693 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{if}\;b \le 2.765077560938384 \cdot 10^{+72}:\\ \;\;\;\;\frac{-c}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

Derivation

  1. Split input into 4 regimes

  2. if b < -1.378869344776828e+154

    1. Initial program 61.0

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

    2. Applied simplify60.9

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

    3. Taylor expanded around -inf 52.3

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{b}{a}}\]

    if -1.378869344776828e+154 < b < 2.732419256347693e-309

    1. Initial program 8.5

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

    2. Applied simplify8.5

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

    4. Applied clear-num8.6

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

    if 2.732419256347693e-309 < b < 2.765077560938384e+72

    1. Initial program 31.1

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

    2. Applied simplify31.1

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

    4. Applied flip--31.1

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

    5. Applied simplify17.0

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

    7. Applied distribute-rgt-neg-out17.0

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

    8. Applied distribute-frac-neg17.0

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

    9. Applied distribute-frac-neg17.0

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

    10. Applied simplify9.1

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

    if 2.765077560938384e+72 < b

    1. Initial program 57.7

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

    2. Applied simplify57.7

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

    4. Applied flip--57.8

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

    5. Applied simplify30.2

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

    7. Applied distribute-rgt-neg-out30.2

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

    8. Applied distribute-frac-neg30.2

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

    9. Applied distribute-frac-neg30.2

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

    10. Applied simplify27.1

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

    11. Taylor expanded around 0 3.4

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

    12. Applied simplify3.4

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

  4. Applied simplify11.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.378869344776828 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{3} \cdot \frac{b}{a}\\ \mathbf{if}\;b \le 2.732419256347693 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{if}\;b \le 2.765077560938384 \cdot 10^{+72}:\\ \;\;\;\;\frac{-c}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))