Average Error: 0.2 → 0.2
Time: 1.0m
Precision: 64
Internal precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
⬇
\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right)\right)\right|\]

Error

Bits error versus x

Derivation

  1. Initial program 0.2

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  2. Applied simplify 0.6

    \[\leadsto \color{blue}{\left|\frac{\frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}{\frac{21}{\left|x\right|}} + \left(\left(\left(\left|x\right| \cdot \frac{2}{3}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\left|x\right| + \left|x\right|\right)\right) + \frac{\left(\left|x\right| \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^3}{5}\right)}{\sqrt{\pi}}\right|}\]

  3. Applied taylor 0.2

    \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right)\right)\right|\]

  4. Taylor expanded around 0 0.2

    \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right)\right)}\right|\]
  5. Removed slow pow expressions

Runtime

Time bar (total: 1.0m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3306700202 760249425 23650823 146866975 2549604806 2890961279)'
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))