Average Error: 44.1 → 0.3
Time: 2.5m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[c \cdot \frac{\frac{3}{-3}}{b + \sqrt{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 44.1

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

  2. Applied simplify44.1

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

  4. Applied flip--44.1

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

  5. Applied simplify0.5

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

  7. Applied frac-2neg0.5

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

  8. Applied simplify0.4

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

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

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

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

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

  12. Applied times-frac0.4

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

  13. Applied times-frac0.4

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

  14. Applied simplify0.4

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

  15. Applied simplify0.3

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

  16. Applied simplify0.3

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +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)))