Average Error: 33.5 → 10.1
Time: 1.8m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -352562.6837860124:\\ \;\;\;\;\frac{\frac{c}{b}}{\frac{3}{\frac{3}{2}}} - \frac{b + b}{a \cdot 3}\\ \mathbf{if}\;b \le 5.2386150073690925 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} + \left(-b\right)}{a}\\ \mathbf{if}\;b \le 7.526428983942289 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 3\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 3\right) \cdot c}} \cdot \left|\sqrt[3]{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right|}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(a \cdot \frac{c}{b}\right) \cdot \frac{3}{2} + \left(\left(-b\right) - b\right))_*}\\ \end{array}\]

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 b < -352562.6837860124

    1. Initial program 30.7

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

    2. Taylor expanded around -inf 12.0

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

    3. Applied simplify8.2

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

    if -352562.6837860124 < b < 5.2386150073690925e-102

    1. Initial program 14.0

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

    3. Applied *-un-lft-identity14.0

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

    4. Applied times-frac14.1

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

    if 5.2386150073690925e-102 < b < 7.526428983942289e-28

    1. Initial program 34.1

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

    3. Applied flip-+34.2

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

    4. Applied simplify17.1

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

    6. Applied add-cube-cbrt17.4

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

    7. Applied sqrt-prod17.4

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

    8. Applied simplify17.4

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

    if 7.526428983942289e-28 < b

    1. Initial program 54.7

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

    3. Applied flip-+54.8

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

    4. Applied simplify26.2

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

    5. Taylor expanded around inf 17.8

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

    6. Applied simplify6.3

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

  4. Applied simplify10.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -352562.6837860124:\\ \;\;\;\;\frac{\frac{c}{b}}{\frac{3}{\frac{3}{2}}} - \frac{b + b}{a \cdot 3}\\ \mathbf{if}\;b \le 5.2386150073690925 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} + \left(-b\right)}{a}\\ \mathbf{if}\;b \le 7.526428983942289 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 3\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 3\right) \cdot c}} \cdot \left|\sqrt[3]{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right|}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(a \cdot \frac{c}{b}\right) \cdot \frac{3}{2} + \left(\left(-b\right) - b\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))