Average Error: 33.2 → 10.6
Time: 1.5m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-\frac{\frac{3}{2}}{b} \le -4.883426180957797 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\ \mathbf{if}\;-\frac{\frac{3}{2}}{b} \le -3.301494844744349 \cdot 10^{-293}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \mathbf{if}\;-\frac{\frac{3}{2}}{b} \le 6.580977947956905 \cdot 10^{-65}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)\right)\\ \end{array}\]

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 (/ (- 3/2) b) < -4.883426180957797e-35

    1. Initial program 28.8

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

    3. Applied flip-+28.9

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

    4. Applied simplify17.7

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

    if -4.883426180957797e-35 < (/ (- 3/2) b) < -3.301494844744349e-293

    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 inf 15.7

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

    3. Applied simplify5.1

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

    if -3.301494844744349e-293 < (/ (- 3/2) b) < 6.580977947956905e-65

    1. Initial program 39.3

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

    2. Taylor expanded around -inf 11.2

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

    if 6.580977947956905e-65 < (/ (- 3/2) b)

    1. Initial program 9.8

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

    3. Applied div-inv9.9

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

  4. Applied simplify10.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-\frac{\frac{3}{2}}{b} \le -4.883426180957797 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\ \mathbf{if}\;-\frac{\frac{3}{2}}{b} \le -3.301494844744349 \cdot 10^{-293}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \mathbf{if}\;-\frac{\frac{3}{2}}{b} \le 6.580977947956905 \cdot 10^{-65}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))