Average Error: 52.1 → 0.2
Time: 1.6m
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\left(-c\right) \cdot \left(3 \cdot a\right)}{3 \cdot a}}{b + \sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.1

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

  2. Initial simplification52.1

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

  4. Applied flip--52.1

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

  5. Applied associate-/l/52.1

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))