Jmat.Real.erf

Average Error: 13.7 → 12.9

Time: 17.4s

Precision: binary64

Math

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 13.7

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

2. Simplified 13.7

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

4. Applied flip3-- 13.7

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

5. Simplified 13.7

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

6. Simplified 13.7

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

8. Applied add-sqr-sqrt 12.9

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

9. Final simplification 12.9

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

Reproduce

herbie shell --seed 2020184 
(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)))))))