Error

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 simplify 1.0

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

3. Applied taylor 1.0

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

4. Taylor expanded around 0 1.0

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

5. Applied simplify 0.9

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

7. Applied pow3 0.8

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

8. Applied pow1 0.8

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

9. Applied pow-flip 0.8

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

10. Applied pow-div 0.8

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

12. Applied add-sqr-sqrt 0.7

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

Runtime

Time bar (total: 2.9m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2983329789 1435411296 742632618 82088948 3754193395 1662918140)'
(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)))))))