Average Error: 33.8 → 7.6
Time: 1.5m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.623859433190926 \cdot 10^{+148}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le -3.242988518190772 \cdot 10^{-273}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}}\\ \mathbf{elif}\;b \le 8.704642209760078 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{c}{3} \cdot 3}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{3} \cdot 3}{\frac{3}{2} \cdot \frac{c \cdot a}{b} - b \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -2.623859433190926e+148

    1. Initial program 59.0

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

    2. Taylor expanded around -inf 2.4

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

    if -2.623859433190926e+148 < b < -3.242988518190772e-273

    1. Initial program 8.6

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

    3. Applied clear-num8.6

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

    if -3.242988518190772e-273 < b < 8.704642209760078e+104

    1. Initial program 31.9

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

    3. Applied flip-+32.0

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

    4. Applied associate-/l/36.3

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

    5. Simplified20.2

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

    7. Applied associate-/r*15.1

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

    9. Applied times-frac9.0

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

    10. Simplified8.9

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

    if 8.704642209760078e+104 < b

    1. Initial program 59.3

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

    3. Applied flip-+59.4

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

    4. Applied associate-/l/59.5

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

    5. Simplified32.8

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

    7. Applied associate-/r*32.2

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

    9. Applied times-frac31.7

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

    10. Simplified31.7

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

    11. Taylor expanded around inf 6.9

      \[\leadsto \frac{\frac{c}{3} \cdot 3}{\color{blue}{\frac{3}{2} \cdot \frac{c \cdot a}{b} - 2 \cdot b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.623859433190926 \cdot 10^{+148}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le -3.242988518190772 \cdot 10^{-273}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}}\\ \mathbf{elif}\;b \le 8.704642209760078 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{c}{3} \cdot 3}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{3} \cdot 3}{\frac{3}{2} \cdot \frac{c \cdot a}{b} - b \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018215 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))