Average Error: 1.5 → 0.8
Time: 7.1m
Precision: 64
Internal Precision: 576
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\frac{\frac{3}{\left|x\right|} + \frac{\frac{4}{8} \cdot 15}{{\left(\left|x\right|\right)}^{3}}}{\left(4 \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^{3}} + \left(\frac{\frac{1}{\left|x\right|}}{2 \cdot \left(\left|x\right| \cdot \left|x\right|\right)} + \frac{1}{\left|x\right|}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around 0 1.5

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

  3. Applied simplify1.0

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

  5. Applied frac-add1.0

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

  6. Applied frac-times1.0

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

  7. Applied simplify0.8

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

  8. Applied simplify0.8

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

Runtime

Time bar (total: 7.1m)Debug logProfile

herbie shell --seed 2018167 
(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)))))))