Average Error: 3.2 → 3.2
Time: 7.1s
Precision: binary64
\[\sqrt{\frac{-4}{3} \cdot p} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-q}{2} \cdot \sqrt{\frac{-27}{{p}^{3}}}\right)\right) - \frac{b}{3} \cdot a\]
↓
\[\sqrt{\frac{-4}{3} \cdot p} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-q}{2} \cdot \sqrt{\frac{-27}{{p}^{3}}}\right)\right) - \frac{b}{3} \cdot a\]

Error

Bits error versus p

Bits error versus q

Bits error versus b

Bits error versus a

Derivation

  1. Initial program 3.2

    \[\sqrt{\frac{-4}{3} \cdot p} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-q}{2} \cdot \sqrt{\frac{-27}{{p}^{3}}}\right)\right) - \frac{b}{3} \cdot a\]

  2. Final simplification3.2

    \[\leadsto \sqrt{\frac{-4}{3} \cdot p} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-q}{2} \cdot \sqrt{\frac{-27}{{p}^{3}}}\right)\right) - \frac{b}{3} \cdot a\]

Reproduce

herbie shell --seed 2020152 
(FPCore (p q b a)
  :name "(- (* (sqrt (* (/ (- 4) 3) p)) (cos (* (/ 1 3) (acos (* (/ (- q) 2) (sqrt (/ (- 27) (pow p 3)))))))) (* (/ b 3) a))"
  :precision binary64
  (- (* (sqrt (* (/ (neg 4.0) 3.0) p)) (cos (* (/ 1.0 3.0) (acos (* (/ (neg q) 2.0) (sqrt (/ (neg 27.0) (pow p 3.0)))))))) (* (/ b 3.0) a)))