Average Error: 33.5 → 9.0
Time: 40.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4290504166653717 \cdot 10^{+119}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b \le 5.1477737191781 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{(\left(-3 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 0.01955255459779226:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(c \cdot a\right) \cdot 3}{\left(\left(-b\right) - \sqrt{(\left(-3 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot 3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

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 < -1.4290504166653717e+119

    1. Initial program 49.9

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

    2. Taylor expanded around -inf 3.7

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

    3. Simplified3.7

      \[\leadsto \color{blue}{(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*}\]

    if -1.4290504166653717e+119 < b < 5.1477737191781e-136

    1. Initial program 11.3

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

    2. Taylor expanded around 0 11.4

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

    3. Simplified11.3

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

    if 5.1477737191781e-136 < b < 0.01955255459779226

    1. Initial program 32.7

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

    2. Taylor expanded around 0 32.7

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

    3. Simplified32.7

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

    5. Applied associate-/r*32.7

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

    7. Applied flip-+32.8

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

    8. Applied associate-/l/32.8

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

    9. Simplified15.6

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

    11. Applied clear-num15.7

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

    if 0.01955255459779226 < b

    1. Initial program 54.6

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

    2. Taylor expanded around 0 54.6

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

    3. Simplified54.6

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

    5. Applied associate-/r*54.6

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

    6. Taylor expanded around inf 6.2

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.4290504166653717 \cdot 10^{+119}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b \le 5.1477737191781 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{(\left(-3 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 0.01955255459779226:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(c \cdot a\right) \cdot 3}{\left(\left(-b\right) - \sqrt{(\left(-3 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot 3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Runtime

Time bar (total: 40.7s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))