Average Error: 28.6 → 0.5
Time: 57.3s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(c \cdot a\right) \cdot 3}{(\left(\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \left(a \cdot -3\right) + \left(a \cdot \left(b \cdot -3\right)\right))_*}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 28.6

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

  3. Applied flip-+28.6

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

  4. Applied associate-/l/28.6

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

  5. Simplified0.6

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

  7. Applied sub-neg0.6

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

  8. Applied distribute-lft-in0.6

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

  9. Simplified0.6

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

  11. Applied *-un-lft-identity0.6

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

  12. Applied associate-/r*0.6

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

  13. Simplified0.5

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

  15. Applied *-commutative0.5

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

  16. Final simplification0.5

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

Runtime

Time bar (total: 57.3s)Debug logProfile

herbie shell --seed 2018256 +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)))