Average Error: 28.5 → 0.4
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 28.5

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

  3. Applied flip-+28.5

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

  4. Applied associate-/l/28.5

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

  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 times-frac0.6

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

  8. Simplified0.5

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

  9. Simplified0.5

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

  11. Applied pow10.5

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

  12. Applied pow10.5

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

  13. Applied pow-prod-down0.5

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

  14. Simplified0.3

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

  16. Applied add-sqr-sqrt0.3

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

  17. Applied sqrt-prod0.4

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

  18. Final simplification0.4

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018348 +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)))