Average Error: 53.0 → 0.2
Time: 54.6s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\sqrt[3]{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_* \cdot \left((\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*\right)}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 53.0

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

  3. Applied flip-+53.0

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

  4. Applied associate-/l/53.0

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

  5. Simplified0.5

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

  7. Applied associate-/r*0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around 0 0.1

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

  11. Applied add-cbrt-cube0.2

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

  12. Final simplification0.2

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

Runtime

Time bar (total: 54.6s)Debug logProfile

herbie shell --seed 2018250 +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)))