Average Error: 1.5 → 0.6
Time: 6.4m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\left(\frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\left(\left|x\right|\right)}^{6}} + \frac{\frac{\frac{\frac{3}{4}}{\left|x\right|}}{\left|x\right|}}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}\right) + \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right|}}{\left|x\right| + \left|x\right|} + \frac{1}{\left|x\right|}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\]

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

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

  3. Taylor expanded around 0 0.7

    \[\leadsto \left(\left(\left(\frac{15}{8} \cdot \frac{1}{\left|x\right|}\right) \cdot \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{6}}} + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{2}\right)\right) + \frac{\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

  4. Applied simplify0.6

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\left(\left|x\right|\right)}^{6}} + \frac{\frac{\frac{\frac{3}{4}}{\left|x\right|}}{\left|x\right|}}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}\right) + \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right|}}{\left|x\right| + \left|x\right|} + \frac{1}{\left|x\right|}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.6

    \[\leadsto \left(\left(\frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\left(\left|x\right|\right)}^{6}} + \frac{\frac{\frac{\frac{3}{4}}{\left|x\right|}}{\left|x\right|}}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}\right) + \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right|}}{\left|x\right| + \left|x\right|} + \frac{1}{\left|x\right|}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}\]
  7. Removed slow pow expressions.

Runtime

Time bar (total: 6.4m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))))))