Average Error: 33.5 → 9.8
Time: 2.8m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.273058773164801 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{\frac{3}{2} \cdot a}{\frac{b}{c}} - \left(b + b\right)}{3 \cdot a}\\ \mathbf{if}\;b \le 1.0696681173689353 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -3.273058773164801e+121

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 10.7

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

    3. Applied simplify3.2

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

    if -3.273058773164801e+121 < b < 1.0696681173689353e-137

    1. Initial program 10.2

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

    3. Applied associate-/r*10.2

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

    4. Applied simplify10.2

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

    if 1.0696681173689353e-137 < b

    1. Initial program 50.2

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

    2. Taylor expanded around inf 22.8

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

    3. Applied simplify11.5

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

  4. Applied simplify9.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -3.273058773164801 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{\frac{3}{2} \cdot a}{\frac{b}{c}} - \left(b + b\right)}{3 \cdot a}\\ \mathbf{if}\;b \le 1.0696681173689353 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))