Average Error: 52.7 → 0.1
Time: 4.7s
Precision: binary64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right) + 0}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

  5. Simplified0.5

    \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right) + 0}{\color{blue}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]
  6. Using strategy rm

  7. Applied clear-num0.6

    \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{3 \cdot \left(a \cdot c\right) + 0}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}}}\]

  8. Simplified0.3

    \[\leadsto \frac{1}{\color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{c}}}\]
  9. Using strategy rm

  10. Applied div-inv0.4

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Simplified0.3

    \[\leadsto \frac{1}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \color{blue}{c}\]
  15. Using strategy rm

  16. Applied associate-*l/0.1

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

  17. Simplified0.1

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

  18. Final simplification0.1

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))