Average Error: 53.7 → 53.7
Time: 9.6s
Precision: binary64
\[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
↓
\[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]

Error

Bits error versus x

Derivation

  1. Initial program 53.7

    \[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]

  2. Final simplification53.7

    \[\leadsto \frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "(- (/ (pow 2 (/ 1 3)) (pow (+ (* (- 3) x) (sqrt (+ 4 (* 9 (pow x 2))))) (/ 1 3))) (/ (pow (+ (* (- 3) x) (sqrt (+ 4 (* 9 (pow x 2))))) (/ 1 3)) (pow 2 (/ 1 3))))"
  :precision binary64
  (- (/ (pow 2.0 (/ 1.0 3.0)) (pow (+ (* (neg 3.0) x) (sqrt (+ 4.0 (* 9.0 (pow x 2.0))))) (/ 1.0 3.0))) (/ (pow (+ (* (neg 3.0) x) (sqrt (+ 4.0 (* 9.0 (pow x 2.0))))) (/ 1.0 3.0)) (pow 2.0 (/ 1.0 3.0)))))