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

Average Error: 37.3 → 37.3

Time: 2.1s

Precision: binary64

Math

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

↓

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

Error

Bits error versus b

Bits error versus a

Bits error versus c

Derivation

1. Initial program 37.3

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

2. Simplified 37.3

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

3. Final simplification 37.3

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

Reproduce

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