Jmat.Real.erf

Average Error: 13.8 → 13.8

Time: 6.8s

Precision: binary64

Math

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

↓

\[\log \left(e^{1 - \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \frac{0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)\right) \cdot 1.0614054289999999\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)\]

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

2. Simplified 13.8

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

4. Applied add-log-exp 13.8

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

5. Applied add-log-exp 13.8

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

6. Applied diff-log 14.6

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

7. Simplified 13.8

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

9. Applied add-cube-cbrt 13.8

   \[\leadsto \log \left(e^{1 - \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \frac{0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \color{blue}{\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)} \cdot 1.0614054289999999\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)\]

10. Final simplification 13.8

    \[\leadsto \log \left(e^{1 - \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \frac{0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)\right) \cdot 1.0614054289999999\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)\]

Reproduce

herbie shell --seed 2020179 
(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)))))))