Average Error: 33.5 → 8.9
Time: 36.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -2.3607951001263578 \cdot 10^{-05}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;-b \le -1.003130952273063 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot a\right) \cdot 3}{3}}{\left(-b\right) - \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}}}{a}\\ \mathbf{elif}\;-b \le 8.382664278578788 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\sqrt{(a \cdot \left(-3 \cdot c\right) + \left(b \cdot b\right))_*} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \end{array}\]

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) < -2.3607951001263578e-05

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 55.2

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    3. Simplified55.2

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

    5. Applied associate-/r*55.2

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

    6. Taylor expanded around inf 5.5

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

    if -2.3607951001263578e-05 < (- b) < -1.003130952273063e-165

    1. Initial program 31.9

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

    2. Taylor expanded around -inf 31.9

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    3. Simplified31.9

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

    5. Applied associate-/r*31.9

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

    7. Applied flip-+32.0

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

    8. Applied associate-/l/32.1

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

    9. Simplified17.8

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

    11. Applied associate-/r*17.8

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

    if -1.003130952273063e-165 < (- b) < 8.382664278578788e+63

    1. Initial program 10.9

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

    2. Taylor expanded around -inf 11.0

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    3. Simplified10.9

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

    5. Applied *-un-lft-identity10.9

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

    6. Applied *-un-lft-identity10.9

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

    7. Applied distribute-lft-out10.9

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

    8. Applied times-frac11.1

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

    9. Simplified11.1

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

    10. Simplified11.0

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

    if 8.382664278578788e+63 < (- b)

    1. Initial program 38.5

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

    2. Taylor expanded around -inf 38.5

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    3. Simplified38.4

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

    4. Taylor expanded around -inf 5.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    5. Simplified5.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;-b \le -2.3607951001263578 \cdot 10^{-05}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;-b \le -1.003130952273063 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot a\right) \cdot 3}{3}}{\left(-b\right) - \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}}}{a}\\ \mathbf{elif}\;-b \le 8.382664278578788 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\sqrt{(a \cdot \left(-3 \cdot c\right) + \left(b \cdot b\right))_*} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 36.7s)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))