Average Error: 52.1 → 0.5
Time: 33.8s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(-c\right) \cdot \left(3 \cdot a\right)}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \left|\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c}\right|}\right)}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Results

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Derivation

  1. Initial program 52.1

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

  2. Initial simplification52.1

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

  4. Applied flip--52.1

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

  5. Applied associate-/l/52.1

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

  6. Simplified0.4

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

  8. Applied add-sqr-sqrt0.5

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied sqrt-prod0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Runtime

Time bar (total: 33.8s)Debug logProfile

herbie shell --seed 2018242 
(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)))