Average Error: 0.2 → 0.1
Time: 56.2s
Precision: 64
Internal Precision: 576
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\sqrt{\frac{1}{\pi}} \cdot (\left({\left(\left|x\right|\right)}^{7}\right) \cdot \frac{1}{21} + \left((\left(\left|x\right|\right) \cdot \left((\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left(\left|x\right|\right) + 2)_*\right) + \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5}\right))_*\right))_*\right|\]

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. Taylor expanded around 0 0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied associate-*r*0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Runtime

Time bar (total: 56.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.10.10.00.10%
herbie shell --seed 2018285 +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)))))))