Average Error: 1.5 → 0.5
Time: 5.3m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[(\left(\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\sqrt{\frac{1}{\pi}}\right) + \left((e^{\log_* (1 + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot (\left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{2}}{\left|x\right|}\right) + 1)_*\right) \cdot \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}}\right))_*)} - 1)^*\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 1.5

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  2. Applied simplify1.2

    \[\leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2 \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*}\]

  3. Taylor expanded around 0 0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2 \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \color{blue}{\left(\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{\left(\left|x\right|\right)}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)})_*\]

  4. Taylor expanded around 0 0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\color{blue}{\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)} \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2 \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{\left(\left|x\right|\right)}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right))_*\]

  5. Applied simplify0.5

    \[\leadsto \color{blue}{(\left(\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\sqrt{\frac{1}{\pi}}\right) + \left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot (\left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{2}}{\left|x\right|}\right) + 1)_*\right) \cdot \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}}\right))_*\right))_*}\]
  6. Using strategy rm

  7. Applied expm1-log1p-u0.5

    \[\leadsto (\left(\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\sqrt{\frac{1}{\pi}}\right) + \color{blue}{\left((e^{\log_* (1 + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot (\left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{2}}{\left|x\right|}\right) + 1)_*\right) \cdot \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}}\right))_*)} - 1)^*\right)})_*\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  (* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x)))))))