Average Error: 52.6 → 0.3
Time: 1.7m
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-2 \cdot c}{\frac{{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}^{3} + {b}^{3}}{\left(b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Results

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Derivation

  1. Initial program 52.6

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

  2. Initial simplification52.6

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

  4. Applied flip--52.6

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

  5. Applied associate-/l/52.6

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.2

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

  9. Taylor expanded around inf 0.1

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

  11. Applied flip3-+0.3

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

  12. Final simplification0.3

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018251 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))