Average Error: 33.6 → 11.2
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6424038809365114 \cdot 10^{-07}:\\ \;\;\;\;\frac{\left(c \cdot \frac{3}{2}\right) \cdot \frac{a}{b} - \left(b + b\right)}{a \cdot 3}\\ \mathbf{if}\;b \le 1.6532775834371295 \cdot 10^{-59}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}\]

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 < -1.6424038809365114e-07

    1. Initial program 31.2

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

    2. Taylor expanded around -inf 12.6

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

    3. Applied simplify8.7

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

    if -1.6424038809365114e-07 < b < 1.6532775834371295e-59

    1. Initial program 15.7

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

    3. Applied div-inv15.8

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

    if 1.6532775834371295e-59 < b

    1. Initial program 53.0

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

    2. Taylor expanded around inf 18.9

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

    3. Applied simplify8.4

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

  4. Applied simplify11.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.6424038809365114 \cdot 10^{-07}:\\ \;\;\;\;\frac{\left(c \cdot \frac{3}{2}\right) \cdot \frac{a}{b} - \left(b + b\right)}{a \cdot 3}\\ \mathbf{if}\;b \le 1.6532775834371295 \cdot 10^{-59}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))