Average Error: 1.5 → 0.5
Time: 5.0m
Precision: 64
Internal Precision: 576
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{15}{8} + \frac{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\right) + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot \left(\frac{1}{\left|x\right|} + \frac{\frac{1}{\left|x\right|}}{2} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)\]

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. Initial simplification1.4

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

  3. Taylor expanded around 0 0.5

    \[\leadsto \left(\frac{1}{\left|x\right|} + \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{\frac{1}{\left|x\right|}}{2}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} + \left(\color{blue}{\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{\left(\left|x\right|\right)}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)} + \frac{\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.5

    \[\leadsto \left(\frac{1}{\left|x\right|} + \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{\frac{1}{\left|x\right|}}{2}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} + \left(\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{\left(\left|x\right|\right)}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

  6. Final simplification0.5

    \[\leadsto \left(\left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{15}{8} + \frac{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\right) + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot \left(\frac{1}{\left|x\right|} + \frac{\frac{1}{\left|x\right|}}{2} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018225 
(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)))))))