Average Error: 1.5 → 0.7
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot \left(\frac{15}{8} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\frac{\sqrt{\pi}}{\frac{1}{\left|x\right|}}} \cdot \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\frac{4}{3}} + \left(\frac{\frac{1}{\left|x\right|}}{2 \cdot \left|x\right|} + 1\right)\right)\]

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 simplify1.5

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

  4. Applied inv-pow1.5

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

  5. Applied pow-pow1.1

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

  6. Applied simplify1.1

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

  7. Taylor expanded around 0 1.1

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

  8. Applied simplify1.1

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

  10. Applied inv-pow1.1

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

  11. Applied pow-pow0.7

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

  12. Applied simplify0.7

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(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)))))))