Average Error: 44.0 → 0.4
Time: 2.4m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{(\left((\left(a \cdot c\right) \cdot 4 + \left(b \cdot b\right))_*\right) \cdot \left(\left(4 \cdot c\right) \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right))_*}}}}{2 \cdot a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 44.0

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

  3. Applied flip-+44.0

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

  4. Applied simplify0.4

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

  6. Applied flip3--0.4

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

  7. Applied simplify0.4

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

Runtime

Time bar (total: 2.4m)Debug logProfile

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