Average Error: 0.2 → 0.2
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\left|x\right|\right) \cdot \left((\left(\left|x\right|\right) \cdot \left(\frac{2}{3} \cdot \left|x\right|\right) + 2)_*\right) + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right))_* \cdot \sqrt{\frac{1}{\pi}} + \sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5}\right)\right|\]

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. Taylor expanded around -inf 0.2

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

  4. Applied distribute-rgt-in0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.20.20.00.20%
herbie shell --seed 2018351 +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)))))))