Average Error: 43.8 → 0.3
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[3 \cdot \frac{\frac{-c}{\sqrt{(\left(-3\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} + b}}{3}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 43.8

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

  2. Applied simplify43.8

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

  4. Applied flip--43.8

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

  5. Applied simplify0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.2

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

  9. Applied times-frac0.4

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

  10. Applied simplify0.3

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))