Average Error: 1.5 → 0.6
Time: 3.3m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left(\left(\sqrt[3]{(\left(\frac{1}{{\left(\left|x\right|\right)}^{\left((2 \cdot 2 + 1)_*\right)}}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*} \cdot \sqrt[3]{(\left((\left(\frac{\frac{1}{2}}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|}\right) + 1)_*\right) \cdot \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}}\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{1}{{\left(\left|x\right|\right)}^{\left((2 \cdot 2 + 1)_*\right)}}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*}\right) + \left(\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \sqrt{\frac{1}{\pi}}\right))_*\]

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 simplify1.2

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

  3. Taylor expanded around 0 0.5

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

  4. Applied simplify0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied simplify0.5

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

  8. Applied simplify0.5

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

  9. Taylor expanded around 0 0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left(\left(\sqrt[3]{(\left(\frac{1}{{\left(\left|x\right|\right)}^{\left((2 \cdot 2 + 1)_*\right)}}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*} \cdot \color{blue}{{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{(\left(\frac{1}{{\left(\left|x\right|\right)}^{\left((2 \cdot 2 + 1)_*\right)}}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*}\right) + \left(\frac{\frac{15}{8}}{\frac{\frac{{\left(\left|x\right|\right)}^{7}}{\sqrt{\frac{1}{\pi}}}}{e^{\left|x\right| \cdot \left|x\right|}}}\right))_*\]

  10. Applied simplify0.6

    \[\leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left(\left(\sqrt[3]{(\left(\frac{1}{{\left(\left|x\right|\right)}^{\left((2 \cdot 2 + 1)_*\right)}}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*} \cdot \sqrt[3]{(\left((\left(\frac{\frac{1}{2}}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|}\right) + 1)_*\right) \cdot \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}}\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{1}{{\left(\left|x\right|\right)}^{\left((2 \cdot 2 + 1)_*\right)}}\right) \cdot \left(\frac{3}{4}\right) + \left((\left(\frac{1}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| + \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*}\right) + \left(\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \sqrt{\frac{1}{\pi}}\right))_*}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(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)))))))