Average Error: 52.5 → 0.4
Time: 5.2s
Precision: binary64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{a \cdot \left(-c\right)}{a \cdot 2}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.5

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

  2. Simplified52.5

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

  4. Applied flip--52.5

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

  5. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{0 - 4 \cdot \left(a \cdot c\right)}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{0 - 4 \cdot \left(a \cdot c\right)}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2}\]
  7. Using strategy rm

  8. Applied sub0-neg0.4

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

  9. Applied distribute-frac-neg0.4

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

  10. Applied distribute-frac-neg0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))