Average Error: 1.5 → 0.7
Time: 2.6m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{\frac{\frac{\frac{-225}{64}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{9}{16}}{{\left(\left|x\right|\right)}^{5} \cdot \left(\frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}\right)}\right) + \frac{1}{\left|x\right|}\right)\]

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. Simplified0.9

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

  3. Taylor expanded around inf 0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}\right)\right) + \frac{1}{\left|x\right|}\right)\]
  4. Using strategy rm

  5. Applied flip-+0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{1}{{\left(\left|x\right|\right)}^{5}} \cdot \color{blue}{\frac{\frac{3}{4} \cdot \frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}}{\frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}}}\right) + \frac{1}{\left|x\right|}\right)\]

  6. Applied frac-times0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \color{blue}{\frac{1 \cdot \left(\frac{3}{4} \cdot \frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}\right)}{{\left(\left|x\right|\right)}^{5} \cdot \left(\frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}\right)}}\right) + \frac{1}{\left|x\right|}\right)\]

  7. Simplified0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{\color{blue}{\frac{\frac{\frac{-225}{64}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{9}{16}}}{{\left(\left|x\right|\right)}^{5} \cdot \left(\frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}\right)}\right) + \frac{1}{\left|x\right|}\right)\]

  8. Final simplification0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{\frac{\frac{\frac{-225}{64}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{9}{16}}{{\left(\left|x\right|\right)}^{5} \cdot \left(\frac{3}{4} - \frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|}\right)}\right) + \frac{1}{\left|x\right|}\right)\]

Reproduce

herbie shell --seed 2019090 
(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)))))))