Average Error: 1.5 → 1.1
Time: 2.5m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)}{\frac{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}}{e^{\left|x\right| \cdot \left|x\right|}}} + \left(\left({\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right) \cdot \left(\left(\frac{15}{8} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \frac{\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

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

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

  4. Applied add-cube-cbrt1.1

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

  6. Applied add-cube-cbrt1.1

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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)))))))