Average Error: 44.2 → 0.4
Time: 1.6m
Precision: 64
Internal Precision: 640
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{3} \cdot \frac{\frac{c}{\frac{1}{3}}}{\left(-b\right) - \sqrt{(\left(c \cdot a\right) \cdot -3 + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 44.2

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

  3. Applied flip-+44.2

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

  4. Applied simplify0.4

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

  5. Applied simplify0.4

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

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

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

  8. Applied times-frac0.6

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

  9. Applied simplify0.6

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

  10. Applied simplify0.4

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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)))