(/ (+ (- b) (sqrt (- (pow b 2) (* 4 ac)))) (* 2 a))

Average Error: 27.5 → 27.4

Time: 2.2s

Precision: binary64

Math

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

↓

\[\frac{\frac{\sqrt{{b}^{2} - 4 \cdot ac} - b}{2}}{a}\]

Error

Bits error versus b

Bits error versus ac

Bits error versus a

Derivation

1. Initial program 27.5

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

2. Simplified 27.4

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{{b}^{2} - 4 \cdot ac} - b}{2}}{a}}\]

3. Final simplification 27.4

   \[\leadsto \frac{\frac{\sqrt{{b}^{2} - 4 \cdot ac} - b}{2}}{a}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (b ac a)
  :name "(/ (+ (- b) (sqrt (- (pow b 2) (* 4 ac)))) (* 2 a))"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (pow b 2.0) (* 4.0 ac)))) (* 2.0 a)))