Average Error: 43.4 → 0.2
Time: 1.1m
Precision: 64
Internal Precision: 640
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot \frac{\frac{-4}{1}}{2}}{\sqrt{\sqrt[3]{{\left((\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right)}^{3}}} + b}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.4

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Applied simplify43.4

    \[\leadsto \color{blue}{\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}}\]
  3. Using strategy rm

  4. Applied flip--43.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}}{2 \cdot a}\]

  5. Applied simplify42.8

    \[\leadsto \frac{\frac{\color{blue}{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_* - b \cdot b}}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}\]

  6. Taylor expanded around 0 0.4

    \[\leadsto \frac{\frac{\color{blue}{-4 \cdot \left(c \cdot a\right)}}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}\]

  7. Applied simplify0.2

    \[\leadsto \color{blue}{\frac{c \cdot \frac{\frac{-4}{1}}{2}}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}\]
  8. Using strategy rm

  9. Applied add-cbrt-cube0.2

    \[\leadsto \frac{c \cdot \frac{\frac{-4}{1}}{2}}{\sqrt{\color{blue}{\sqrt[3]{\left((\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_* \cdot (\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}} + b}\]

  10. Applied simplify0.2

    \[\leadsto \frac{c \cdot \frac{\frac{-4}{1}}{2}}{\sqrt{\sqrt[3]{\color{blue}{{\left((\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right)}^{3}}}} + b}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))