Average Error: 44.2 → 11.7

Time: 5.0m

Precision: 64

Internal Precision: 128

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

↓

\[-\frac{c}{b}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

Derivation

Initial program 44.2
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Simplified44.2
\[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]
Taylor expanded around inf 11.7
\[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
Simplified11.7
\[\leadsto \color{blue}{-\frac{c}{b}}\]
Final simplification11.7
\[\leadsto -\frac{c}{b}\]

Reproduce

herbie shell --seed 2019088 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))