Average Error: 32.9 → 14.7
Time: 34.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6812793600290991 \cdot 10^{+112}:\\ \;\;\;\;\frac{(-2 \cdot b + \left(\left(a \cdot c\right) \cdot \frac{\frac{3}{2}}{b}\right))_*}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.7911706408590685 \cdot 10^{-63}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]

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 b < -1.6812793600290991e+112

    1. Initial program 47.4

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

    2. Taylor expanded around -inf 9.2

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

    3. Simplified9.2

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

    if -1.6812793600290991e+112 < b < 4.7911706408590685e-63

    1. Initial program 12.3

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

    3. Applied add-sqr-sqrt12.5

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

    if 4.7911706408590685e-63 < b

    1. Initial program 53.0

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

    2. Taylor expanded around inf 19.5

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

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6812793600290991 \cdot 10^{+112}:\\ \;\;\;\;\frac{(-2 \cdot b + \left(\left(a \cdot c\right) \cdot \frac{\frac{3}{2}}{b}\right))_*}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.7911706408590685 \cdot 10^{-63}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]

Runtime

Time bar (total: 34.1s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))