Average Error: 52.5 → 0.1
Time: 26.1s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{\frac{{\left(\left(a \cdot c\right) \cdot -3\right)}^{3} + {\left(b \cdot b\right)}^{3}}{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)\right) + \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}}}\]

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 52.5

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

  2. Initial simplification52.5

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

  4. Applied flip--52.5

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

  5. Applied associate-/l/52.5

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

  6. Simplified0.5

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

  8. Applied associate-/r*0.4

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

  9. Taylor expanded around -inf 0.1

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

  10. Simplified0.1

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

  12. Applied flip3-+0.1

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

  13. Final simplification0.1

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

Runtime

Time bar (total: 26.1s)Debug logProfile

herbie shell --seed 2018248 
(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)))