Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 5

Speedup: 1.2×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt (PI))) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt (PI))) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot t\_0\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \left(-t\_0\right)\right)\right) + \frac{15}{8} \cdot {x}^{-7}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (/ (pow (exp x) x) (sqrt (PI)))
    (+
     (+
      (+ t_0 (* (/ 1.0 2.0) (* (/ 1.0 (* x x)) t_0)))
      (* (/ 3.0 4.0) (* (/ -1.0 (* (* x x) (* x x))) (- t_0))))
     (* (/ 15.0 8.0) (pow x -7.0))))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. *-lft-identity N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\color{blue}{\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

6. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \color{blue}{{x}^{-7}}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\color{blue}{\frac{1}{x \cdot x}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot {x}^{-7}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 100.0

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 100.0

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

   11. lift-fabs.f64 N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

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

   16. unpow1 100.0

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

   17. lift-fabs.f64 N/A

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

   18. rem-sqrt-square-rev N/A

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 100.0

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \left(-\frac{1}{\left|x\right|}\right)\right)\right) + \frac{15}{8} \cdot {x}^{-7}\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot t\_0\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \left(-t\_0\right)\right)\right) + {x}^{-7} \cdot 1.875\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (* (/ 1.0 (sqrt (PI))) (exp (* x x)))
    (+
     (+
      (+ t_0 (* (/ 1.0 2.0) (* (/ 1.0 (* x x)) t_0)))
      (* (/ 3.0 4.0) (* (/ -1.0 (* (* x x) (* x x))) (- t_0))))
     (* (pow x -7.0) 1.875)))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. *-lft-identity N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\color{blue}{\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

6. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \color{blue}{{x}^{-7}}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\color{blue}{\frac{1}{x \cdot x}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot {x}^{-7}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-/.f64 N/A

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

   5. metadata-eval 100.0

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + {x}^{-7} \cdot \color{blue}{1.875}\right) \]

10. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \color{blue}{{x}^{-7} \cdot 1.875}\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \left(-\frac{1}{\left|x\right|}\right)\right)\right) + {x}^{-7} \cdot 1.875\right) \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \frac{\left(\frac{0.75}{{x}^{4}} + 1\right) + \frac{0.5}{x \cdot x}}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt (PI))) (exp (* x x)))
  (/ (+ (+ (/ 0.75 (pow x 4.0)) 1.0) (/ 0.5 (* x x))) (fabs x))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. *-lft-identity N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\color{blue}{\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-add-rev N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-abs-rev N/A

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

   13. lower-*.f64 N/A

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

   14. lower-fabs.f64 99.7

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\color{blue}{\left|x\right|}} + \frac{15}{8} \cdot {x}^{-7}\right) \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}} + \frac{15}{8} \cdot {x}^{-7}\right) \]

10. Taylor expanded in x around inf

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{{x}^{4} \cdot \left|x\right|}\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

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

    2. metadata-eval N/A

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

    3. unpow3 N/A

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

    4. sqr-abs-rev N/A

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

    5. unpow2 N/A

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. +-commutative N/A

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

    9. associate-+r+ N/A

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

    10. associate-/r* N/A

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

    11. associate-/r* N/A

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

    12. div-add-rev N/A

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

    13. div-add-rev N/A

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

12. Applied rewrites 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{\left(\frac{0.75}{{x}^{4}} + 1\right) + \frac{0.5}{x \cdot x}}{\left|x\right|}} \]

13. Final simplification 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \frac{\left(\frac{0.75}{{x}^{4}} + 1\right) + \frac{0.5}{x \cdot x}}{\left|x\right|} \]
14. Add Preprocessing

Alternative 4: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + {x}^{-7} \cdot 1.875\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt (PI))) (exp (* x x)))
  (+ (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x)) (* (pow x -7.0) 1.875))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. *-lft-identity N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\color{blue}{\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-add-rev N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-abs-rev N/A

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

   13. lower-*.f64 N/A

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

   14. lower-fabs.f64 99.7

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\color{blue}{\left|x\right|}} + \frac{15}{8} \cdot {x}^{-7}\right) \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}} + \frac{15}{8} \cdot {x}^{-7}\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 99.7

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \color{blue}{{x}^{-7} \cdot \frac{15}{8}}\right) \]

    4. lift-/.f64 N/A

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

    5. metadata-eval 99.7

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + {x}^{-7} \cdot \color{blue}{1.875}\right) \]

11. Applied rewrites 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \color{blue}{{x}^{-7} \cdot 1.875}\right) \]

12. Final simplification 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + {x}^{-7} \cdot 1.875\right) \]
13. Add Preprocessing

Alternative 5: 99.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt (PI))) (exp (* x x)))
  (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. *-lft-identity N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\color{blue}{\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-add-rev N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-abs-rev N/A

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

   13. lower-*.f64 N/A

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

   14. lower-fabs.f64 99.7

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\color{blue}{\left|x\right|}} + \frac{15}{8} \cdot {x}^{-7}\right) \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}} + \frac{15}{8} \cdot {x}^{-7}\right) \]

10. Taylor expanded in x around inf

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{\frac{1}{2} \cdot 1}{{\left(\left|x\right|\right)}^{3}}}\right) \]

    2. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \frac{\color{blue}{\frac{1}{2}}}{{\left(\left|x\right|\right)}^{3}}\right) \]

    3. unpow3 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \frac{\frac{1}{2}}{\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}}\right) \]

    4. sqr-abs-rev N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \frac{\frac{1}{2}}{\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|}\right) \]

    5. unpow2 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \frac{\frac{1}{2}}{\color{blue}{{x}^{2}} \cdot \left|x\right|}\right) \]

    6. associate-/r* N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{\frac{\frac{1}{2}}{{x}^{2}}}{\left|x\right|}}\right) \]

    7. div-add-rev N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1 + \frac{\frac{1}{2}}{{x}^{2}}}{\left|x\right|}} \]

    8. +-commutative N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{{x}^{2}} + 1}}{\left|x\right|} \]

    9. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{\frac{\frac{1}{2}}{{x}^{2}} + 1}{\left|x\right|}} \]

    10. lower-+.f64 N/A

        \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{{x}^{2}} + 1}}{\left|x\right|} \]

    11. lower-/.f64 N/A

        \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{{x}^{2}}} + 1}{\left|x\right|} \]

    12. unpow2 N/A

        \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\frac{\frac{1}{2}}{\color{blue}{x \cdot x}} + 1}{\left|x\right|} \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\frac{\frac{1}{2}}{\color{blue}{x \cdot x}} + 1}{\left|x\right|} \]

    14. lower-fabs.f64 99.7

        \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\frac{0.5}{x \cdot x} + 1}{\color{blue}{\left|x\right|}} \]

12. Applied rewrites 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}} \]

13. Final simplification 99.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025006 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt (PI))) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))