Jmat.Real.erf

Average Error: 14.0 → 14.0

Time: 8.8s

Precision: binary64

Math

\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)\]

Error

Bits error versus x

Derivation

1. Initial program 14.0

   \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied flip-+ 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)}{1 - 0.32759110000000002 \cdot \left|x\right|}}} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Applied associate-/r/ 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \color{blue}{\left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right)} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

5. Applied associate-*l* 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \color{blue}{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
6. Using strategy rm

7. Applied distribute-lft-in 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \color{blue}{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

8. Applied associate-+r+ 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -0.284496735999999972\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

9. Simplified 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\color{blue}{\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)} + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
10. Using strategy rm

11. Applied add-log-exp 14.0

    \[\leadsto 1 - \color{blue}{\log \left(e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

12. Applied add-log-exp 14.0

    \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)\]

13. Applied diff-log 14.0

    \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\right)}\]

14. Simplified 14.0

    \[\leadsto \log \color{blue}{\left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

15. Final simplification 14.0

    \[\leadsto \log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (neg (* (fabs x) (fabs x)))))))