Average Error: 1.5 → 0.8
Time: 2.6m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\left(\frac{\frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\left(\left|x\right|\right)}^{3}}}{{\left(\left|x\right|\right)}^{3}} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{\left(2 + 3\right)}}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

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. Applied simplify1.1

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

  3. Taylor expanded around 0 1.0

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

  4. Applied simplify1.0

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

  5. Taylor expanded around 0 1.0

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

  6. Applied simplify0.8

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

Runtime

Time bar (total: 2.6m)Debug log

herbie shell --seed '#(2151728608 3767143998 4065594768 704903143 4284770345 1704475780)' 
(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)))))))