Average Error: 0.2 → 0.2
Time: 5.1m
Precision: 64
Internal Precision: 576
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{1}{5}\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left((\left(\left|x\right| \cdot \frac{2}{3}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot {\left(\left|x\right| \cdot \left|x\right|\right)}^{3}\right))_*\right|\]

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. Initial simplification0.2

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

  3. Final simplification0.2

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

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed 2018217 +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)))))))