Average Error: 33.9 → 6.7
Time: 2.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.402873155637692 \cdot 10^{+133}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{if}\;b \le -6.473307266256287 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{3}}{a}\\ \mathbf{if}\;b \le 7.400171249158961 \cdot 10^{+124}:\\ \;\;\;\;\frac{c}{3} \cdot \frac{3}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(\frac{3}{2} \cdot c\right) \cdot \frac{a}{b} - b \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 4 regimes

  2. if b < -6.402873155637692e+133

    1. Initial program 54.4

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

    2. Taylor expanded around -inf 3.7

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

    if -6.402873155637692e+133 < b < -6.473307266256287e-217

    1. Initial program 8.1

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

    3. Applied associate-/r*8.1

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

    4. Applied simplify8.1

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

    if -6.473307266256287e-217 < b < 7.400171249158961e+124

    1. Initial program 30.4

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

    3. Applied flip-+30.5

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

    4. Applied simplify16.1

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

    6. Applied *-un-lft-identity16.1

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

    7. Applied times-frac15.4

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

    8. Applied times-frac11.2

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

    9. Applied simplify11.2

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

    10. Applied simplify9.5

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

    if 7.400171249158961e+124 < b

    1. Initial program 60.3

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

    3. Applied flip-+60.3

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

    4. Applied simplify35.8

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

    6. Applied *-un-lft-identity35.8

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

    7. Applied times-frac36.7

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

    8. Applied times-frac36.3

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

    9. Applied simplify36.3

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

    10. Applied simplify34.6

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

    11. Taylor expanded around inf 6.6

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

    12. Applied simplify1.8

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))