Average Error: 33.3 → 8.8
Time: 1.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.705432594986331 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\frac{3}{2} \cdot a}{\frac{b}{c}} - 2 \cdot b}{a \cdot 3}\\ \mathbf{if}\;b \le 1.2033851405978159 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a} \cdot \frac{1}{3}\\ \mathbf{if}\;b \le 1.521253697457228 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 3\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 4 regimes

  2. if b < -1.705432594986331e+129

    1. Initial program 52.0

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

    3. Applied *-un-lft-identity52.0

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

    4. Applied times-frac52.0

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

    5. Applied simplify52.0

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

    6. Taylor expanded around -inf 10.8

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

    7. Applied simplify3.3

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

    if -1.705432594986331e+129 < b < 1.2033851405978159e-276

    1. Initial program 9.3

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

    3. Applied *-un-lft-identity9.3

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

    4. Applied times-frac9.4

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

    5. Applied simplify9.4

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

    if 1.2033851405978159e-276 < b < 1.521253697457228e+23

    1. Initial program 27.4

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

    3. Applied flip-+27.5

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

    4. Applied simplify16.6

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

    if 1.521253697457228e+23 < b

    1. Initial program 55.4

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

    2. Taylor expanded around inf 17.8

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

    3. Applied simplify5.2

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

  4. Applied simplify8.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.705432594986331 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\frac{3}{2} \cdot a}{\frac{b}{c}} - 2 \cdot b}{a \cdot 3}\\ \mathbf{if}\;b \le 1.2033851405978159 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a} \cdot \frac{1}{3}\\ \mathbf{if}\;b \le 1.521253697457228 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 3\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))