Average Error: 33.9 → 13.1
Time: 1.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.719140003361598 \cdot 10^{-280}:\\ \;\;\;\;\left(\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{if}\;b \le 4.324045525595186 \cdot 10^{+119}:\\ \;\;\;\;\frac{-c}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -4.719140003361598e-280

    1. Initial program 21.2

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

    2. Applied simplify21.2

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

    4. Applied div-inv21.2

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

    if -4.719140003361598e-280 < b < 4.324045525595186e+119

    1. Initial program 33.2

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

    2. Applied simplify33.2

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

    4. Applied flip--33.3

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

    5. Applied simplify17.3

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

    7. Applied distribute-rgt-neg-out17.3

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

    8. Applied distribute-frac-neg17.3

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

    9. Applied distribute-frac-neg17.3

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

    10. Applied simplify9.8

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

    if 4.324045525595186e+119 < b

    1. Initial program 59.7

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

    2. Applied simplify59.8

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

    4. Applied flip--59.8

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

    5. Applied simplify34.1

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

    7. Applied distribute-rgt-neg-out34.1

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

    8. Applied distribute-frac-neg34.1

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

    9. Applied distribute-frac-neg34.1

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

    10. Applied simplify32.5

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

    11. Taylor expanded around 0 2.2

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

    12. Applied simplify2.2

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

  4. Applied simplify13.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -4.719140003361598 \cdot 10^{-280}:\\ \;\;\;\;\left(\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{if}\;b \le 4.324045525595186 \cdot 10^{+119}:\\ \;\;\;\;\frac{-c}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))