Average Error: 33.0 → 10.2
Time: 1.5m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-2}{3} \cdot b \le -1.8700313055846637 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\frac{-3}{2}}{3}}{\frac{b}{c}}\\ \mathbf{if}\;\frac{-2}{3} \cdot b \le 1.5502720957942897 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{if}\;\frac{-2}{3} \cdot b \le 1.2658088817796178 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{-2}{3} \cdot b}{a}\\ \mathbf{if}\;\frac{-2}{3} \cdot b \le 6.383327851614631 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 5 regimes

  2. if (* -2/3 b) < -1.8700313055846637e-14

    1. Initial program 54.0

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

    3. Applied associate-/r*54.0

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

    4. Taylor expanded around inf 17.4

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

    5. Applied simplify7.5

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

    if -1.8700313055846637e-14 < (* -2/3 b) < 1.5502720957942897e-183

    1. Initial program 21.4

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

    3. Applied flip-+21.7

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

    4. Applied simplify16.3

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

    if 1.5502720957942897e-183 < (* -2/3 b) < 1.2658088817796178e-140

    1. Initial program 15.8

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

    3. Applied associate-/r*15.9

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

    4. Taylor expanded around -inf 46.7

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

    if 1.2658088817796178e-140 < (* -2/3 b) < 6.383327851614631e+142

    1. Initial program 6.0

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

    3. Applied associate-/r*6.0

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

    if 6.383327851614631e+142 < (* -2/3 b)

    1. Initial program 55.9

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

    3. Applied div-inv55.9

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

    4. Taylor expanded around -inf 3.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))