Average Error: 44.1 → 0.4
Time: 1.8m
Precision: 64
Internal Precision: 640
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\left(\frac{1}{c} \cdot \frac{a}{-a}\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} + b\right)}\]

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

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

  4. Applied flip--44.1

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

  5. Applied simplify0.5

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

  7. Applied clear-num0.6

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

  8. Applied simplify0.4

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(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)))