Average Error: 1.5 → 1.0
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + \frac{\frac{3}{4}}{\left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} + \left(\frac{1}{\left|x\right| \cdot \sqrt{\pi}} + \frac{\frac{1}{{\left(\left|x\right|\right)}^{5}}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{\frac{15}{8}}{\sqrt{\pi}}}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|}\]

Error

Bits error versus x

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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

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

  3. Simplified1.3

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

  4. Simplified1.3

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

  5. Taylor expanded around -inf 1.0

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

  6. Final simplification1.0

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018254 
(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)))))))