Average Error: 0.2 → 0.2
Time: 3.6m
Precision: 64
Internal Precision: 384
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]

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 simplify0.6

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

  3. Taylor expanded around 0 0.2

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

Runtime

Time bar (total: 3.6m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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)))))))