Average Error: 28.8 → 0.5
Time: 2.5m
Precision: 64
Internal Precision: 640
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(c \cdot \left(a \cdot 3\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}{3 \cdot a}\]

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 flip-+28.8

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

  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 div-inv0.5

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

Runtime

Time bar (total: 2.5m)Debug logProfile

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