Average Error: 33.6 → 7.6
Time: 2.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.344987796431683 \cdot 10^{+150}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 4.728715747220183 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}\\ \mathbf{elif}\;b \le 2.2043707357501494 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \left(b - \frac{c \cdot a}{b} \cdot \frac{3}{2}\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -7.344987796431683e+150

    1. Initial program 60.3

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

    2. Taylor expanded around -inf 2.7

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}}\]

    if -7.344987796431683e+150 < b < 4.728715747220183e-304

    1. Initial program 8.5

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

    3. Applied clear-num8.5

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

    if 4.728715747220183e-304 < b < 2.2043707357501494e+150

    1. Initial program 34.8

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

    3. Applied flip-+34.9

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

    4. Applied associate-/l/39.1

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

    5. Simplified20.9

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

    7. Applied associate-/r*15.3

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

    9. Applied times-frac15.1

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

    10. Simplified15.1

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

    11. Simplified8.9

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

    if 2.2043707357501494e+150 < b

    1. Initial program 62.5

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

    3. Applied flip-+62.5

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

    4. Applied associate-/l/62.5

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

    5. Simplified36.4

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

    7. Applied associate-/r*36.2

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

    9. Applied times-frac36.2

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

    10. Simplified36.2

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

    11. Simplified36.1

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

    12. Taylor expanded around inf 6.2

      \[\leadsto \frac{1 \cdot c}{\left(-b\right) - \color{blue}{\left(b - \frac{3}{2} \cdot \frac{a \cdot c}{b}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.344987796431683 \cdot 10^{+150}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 4.728715747220183 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}\\ \mathbf{elif}\;b \le 2.2043707357501494 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \left(b - \frac{c \cdot a}{b} \cdot \frac{3}{2}\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))