Average Error: 43.9 → 0.4
Time: 2.3m
Precision: 64
Internal Precision: 640
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{(\left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + b)_*}}{2 \cdot a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.9

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

  2. Applied simplify43.9

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

  4. Applied flip--44.0

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

  5. Applied simplify0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied sqrt-prod0.5

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

  9. Applied fma-def0.4

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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)))