Average Error: 52.7 → 0.1
Time: 1.3m
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{4}{2} \cdot \left(-c\right)}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  2. Initial simplification52.7

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

  4. Applied flip--52.7

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

  5. Applied associate-/l/52.7

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

  6. Simplified0.4

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

  8. Applied distribute-lft-neg-out0.4

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

  9. Applied distribute-frac-neg0.4

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))