Average Error: 53.0 → 0.3
Time: 1.7m
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-1}{(\left(\frac{1}{c}\right) \cdot \left(\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \left(\frac{b}{c}\right))_*}\]

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 simplify0.4

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

  6. Applied clear-num0.4

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

  7. Applied simplify0.4

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

  9. Applied frac-2neg0.4

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

  10. Applied simplify0.3

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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)))