Average Error: 33.2 → 12.5
Time: 2.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le -5.1266461288916434 \cdot 10^{+306}:\\ \;\;\;\;\frac{-c}{b + b}\\ \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le -8.10967986457994 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le 1.2189048961835399 \cdot 10^{-219}:\\ \;\;\;\;\frac{-c}{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le 7.233719411252099 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b + b}\\ \end{array}\]

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 (/ 1 (/ (* 3 a) (exp (log (- (sqrt (fma (* 3 a) (- c) (* b b))) b))))) < -5.1266461288916434e+306 or 7.233719411252099e+307 < (/ 1 (/ (* 3 a) (exp (log (- (sqrt (fma (* 3 a) (- c) (* b b))) b)))))

    1. Initial program 61.7

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

    2. Applied simplify61.7

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

    4. Applied flip--62.3

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

    5. Applied simplify49.5

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

    6. Taylor expanded around 0 39.0

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

    7. Applied simplify31.0

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

    if -5.1266461288916434e+306 < (/ 1 (/ (* 3 a) (exp (log (- (sqrt (fma (* 3 a) (- c) (* b b))) b))))) < -8.10967986457994e-231 or 1.2189048961835399e-219 < (/ 1 (/ (* 3 a) (exp (log (- (sqrt (fma (* 3 a) (- c) (* b b))) b))))) < 7.233719411252099e+307

    1. Initial program 2.6

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

    2. Applied simplify2.6

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

    4. Applied clear-num2.7

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

    if -8.10967986457994e-231 < (/ 1 (/ (* 3 a) (exp (log (- (sqrt (fma (* 3 a) (- c) (* b b))) b))))) < 1.2189048961835399e-219

    1. Initial program 50.4

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

    2. Applied simplify50.4

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

    4. Applied flip--50.8

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

    5. Applied simplify15.4

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

    7. Applied distribute-rgt-neg-out15.4

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

    8. Applied distribute-frac-neg15.4

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

    9. Applied distribute-frac-neg15.4

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

    10. Applied simplify0.5

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

  4. Applied simplify12.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le -5.1266461288916434 \cdot 10^{+306}:\\ \;\;\;\;\frac{-c}{b + b}\\ \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le -8.10967986457994 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le 1.2189048961835399 \cdot 10^{-219}:\\ \;\;\;\;\frac{-c}{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{if}\;\frac{1}{\frac{a \cdot 3}{e^{\log \left(\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}} \le 7.233719411252099 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b + b}\\ \end{array}}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018207 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))