Average Error: 43.7 → 42.6
Time: 35.5s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\sqrt{\left({\left((\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt[3]{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \left(\sqrt[3]{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}\right)}\right) \cdot \left(\sqrt[3]{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right)} - b}{3 \cdot a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 43.7

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

  2. Initial simplification43.7

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

  4. Applied add-cube-cbrt44.1

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

  6. Applied pow1/342.6

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

  8. Applied add-cbrt-cube42.6

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

  10. Applied add-sqr-sqrt42.6

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

  11. Applied cbrt-prod42.6

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

  12. Final simplification42.6

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

Runtime

Time bar (total: 35.5s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(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)))