Average Error: 0.2 → 0.2
Time: 50.9s
Precision: 64
Internal Precision: 576
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{\left|x\right|}{5}\right) \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + \left((\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{\left|x\right|}{\frac{3}{2}}\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\frac{\frac{\left|x\right|}{\sqrt{\pi}}}{21} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)\right))_*\right|\]

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 simplify0.2

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

  3. Taylor expanded around 0 0.2

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

  4. Applied simplify0.2

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

Runtime

Time bar (total: 50.9s)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(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)))))))