Average Error: 52.7 → 0.2
Time: 46.9s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{c}{-3} \cdot 3}{b + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}\]

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 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. Applied associate-/l/52.7

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

  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 sub-neg0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Simplified0.5

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

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

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

  12. Applied times-frac0.5

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

  13. Simplified0.5

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

  14. Simplified0.4

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

  16. Applied associate-*r/0.2

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

  17. Final simplification0.2

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

Runtime

Time bar (total: 46.9s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))