Average Error: 1.5 → 0.8
Time: 8.9m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\left(\frac{1}{\left|x\right|} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|} + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{1}{\left|x\right|}}{\frac{4}{3} \cdot \left|x\right|}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{1}{\left|x\right|}}} + \frac{\frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\left(\left|x\right|\right)}^{3}}}{{\left(\left|x\right|\right)}^{3}}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

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 simplify1.1

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

  3. Taylor expanded around 0 1.1

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

  4. Applied simplify0.9

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

  6. Applied *-un-lft-identity0.9

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

  7. Applied add-sqr-sqrt0.9

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

  8. Applied times-frac1.0

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

  9. Applied associate-/l*1.0

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

  10. Applied simplify0.8

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

  11. Taylor expanded around 0 0.8

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

  12. Applied simplify0.8

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

  13. Applied simplify0.8

    \[\leadsto \color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|} + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{1}{\left|x\right|}}{\frac{4}{3} \cdot \left|x\right|}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{1}{\left|x\right|}}} + \frac{\frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\left(\left|x\right|\right)}^{3}}}{{\left(\left|x\right|\right)}^{3}}\right)\right)} \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]
  14. Removed slow pow expressions.

Runtime

Time bar (total: 8.9m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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)))))))