Average Error: 36.4 → 36.4
Time: 6.2s
Precision: binary64
\[\frac{-b}{3} + 2 \cdot \left(\sqrt{-q} \cdot \cos \left(\cos^{-1} \left(\frac{r}{\left(\left(3 \cdot \sqrt{-q}\right) \cdot \sqrt{-q}\right) \cdot \sqrt{-q}}\right)\right)\right)\]
↓
\[2 \cdot \left(\sqrt{-q} \cdot \cos \left(\cos^{-1} \left(\frac{r}{\left(\left(3 \cdot \sqrt{-q}\right) \cdot \sqrt{-q}\right) \cdot \sqrt{-q}}\right)\right)\right) - \frac{b}{3}\]

Error

Bits error versus b

Bits error versus q

Bits error versus r

Derivation

  1. Initial program 36.4

    \[\frac{-b}{3} + 2 \cdot \left(\sqrt{-q} \cdot \cos \left(\cos^{-1} \left(\frac{r}{\left(\left(3 \cdot \sqrt{-q}\right) \cdot \sqrt{-q}\right) \cdot \sqrt{-q}}\right)\right)\right)\]

  2. Simplified36.4

    \[\leadsto \color{blue}{2 \cdot \left(\sqrt{-q} \cdot \cos \left(\cos^{-1} \left(\frac{r}{\left(\left(3 \cdot \sqrt{-q}\right) \cdot \sqrt{-q}\right) \cdot \sqrt{-q}}\right)\right)\right) - \frac{b}{3}}\]

  3. Final simplification36.4

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

Reproduce

herbie shell --seed 2020153 
(FPCore (b q r)
  :name "(+ (/ (- b) 3) (* 2 (* (sqrt (- q)) (cos (acos (/ r (* (* (* 3 (sqrt (- q))) (sqrt (- q))) (sqrt (- q)))))))))"
  :precision binary64
  (+ (/ (neg b) 3.0) (* 2.0 (* (sqrt (neg q)) (cos (acos (/ r (* (* (* 3.0 (sqrt (neg q))) (sqrt (neg q))) (sqrt (neg q))))))))))