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

Average Error: 34.2 → 34.2

Time: 2.3s

Precision: binary64

Math

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

↓

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

Error

Bits error versus b

Bits error versus a

Bits error versus c

Derivation

1. Initial program 34.2

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

2. Simplified 34.2

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

3. Final simplification 34.2

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

Reproduce

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