Average Error: 43.7 → 0.4
Time: 43.2s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3}{b \cdot \frac{-3}{c} + \frac{\sqrt{\left(c \cdot -3\right) \cdot a + b \cdot b}}{\frac{c}{-3}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 43.7

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

  3. Applied flip-+43.7

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

  4. Applied associate-/l/43.7

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

  5. Simplified0.5

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

  7. Applied associate-/l*0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied associate-/r*0.5

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

  11. Simplified0.4

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

  13. Applied distribute-rgt-in0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Runtime

Time bar (total: 43.2s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))