Average Error: 28.7 → 0.5
Time: 3.7m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot \left(4 \cdot a\right) + \left(b \cdot b - b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\sqrt[3]{\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right) \cdot \left(\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right) \cdot \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)\right)}}\right) \cdot \left(2 \cdot a\right)}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.7

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

  3. Applied flip-+28.6

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

  4. Applied associate-/l/28.6

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

  5. Simplified0.4

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

  7. Applied add-cbrt-cube0.5

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

  8. Final simplification0.5

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))