Average Error: 52.7 → 51.9
Time: 21.2s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\sqrt[3]{(\left(\sqrt{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_* \cdot \left((\left(\sqrt{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_* \cdot (\left(\sqrt{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*\right)}}{3 \cdot a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.7

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

  2. Initial simplification52.7

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

  4. Applied add-sqr-sqrt52.5

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

  5. Applied fma-neg51.9

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

  7. Applied add-cbrt-cube51.9

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

  8. Final simplification51.9

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

Runtime

Time bar (total: 21.2s)Debug logProfile

herbie shell --seed 2018227 +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)))