Average Error: 1.5 → 0.5
Time: 8.8m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)}{{\left(\left|x\right|\right)}^{7}}\right) + \frac{\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{4}} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\frac{\frac{\left|x\right|}{\sqrt{\frac{1}{\pi}}}}{e^{\left|x\right| \cdot \left|x\right|}}}\]

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

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

  3. Taylor expanded around 0 0.5

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

  4. Taylor expanded around 0 0.5

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

  5. Applied simplify0.5

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

  7. Applied associate-*l/0.5

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

Runtime

Time bar (total: 8.8m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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)))))))