Average Error: 28.8 → 28.7
Time: 34.8s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(-b\right) + \sqrt{(b \cdot b + \left(-\sqrt[3]{{\left(\left(3 \cdot c\right) \cdot a\right)}^{3}}\right))_* + 0}}{3 \cdot a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 28.8

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

  3. Applied prod-diff28.7

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

  4. Applied simplify28.7

    \[\leadsto \frac{\left(-b\right) + \sqrt{(b \cdot b + \left(-c \cdot \left(3 \cdot a\right)\right))_* + \color{blue}{0}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube28.7

    \[\leadsto \frac{\left(-b\right) + \sqrt{(b \cdot b + \left(-\color{blue}{\sqrt[3]{\left(\left(c \cdot \left(3 \cdot a\right)\right) \cdot \left(c \cdot \left(3 \cdot a\right)\right)\right) \cdot \left(c \cdot \left(3 \cdot a\right)\right)}}\right))_* + 0}}{3 \cdot a}\]

  7. Applied simplify28.7

    \[\leadsto \frac{\left(-b\right) + \sqrt{(b \cdot b + \left(-\sqrt[3]{\color{blue}{{\left(\left(3 \cdot c\right) \cdot a\right)}^{3}}}\right))_* + 0}}{3 \cdot a}\]

Runtime

Time bar (total: 34.8s)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))