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

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

Alternatives: 12

Speedup: 1.1×

• Could not converge on a ground truth

  1. x = 8.014930728106168e+80

• 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{x}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (pow (sqrt (PI)) -1.0) (pow (exp (- x)) (- x)))
  (-
   (/ (+ (+ (/ 0.75 (pow x 4.0)) (/ 0.5 (* x x))) 1.0) (fabs x))
   (/ -1.875 (pow x 7.0)))))

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right)} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs-rev N/A

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

   6. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{-x}\right)}^{\color{blue}{\left(-x\right)}}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{{\left(e^{-x}\right)}^{\left(-x\right)}}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]
10. Step-by-step derivation

11. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{\color{blue}{{x}^{7}}}\right) \]

12. Final simplification 100.0%

    \[\leadsto \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{x}^{7}}\right) \]
13. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{-1}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(t\_0 + {2}^{-1} \cdot t\_1\right) + \frac{3}{4} \cdot \left(\left(t\_1 \cdot t\_0\right) \cdot t\_0\right)\right) + \frac{15}{8} \cdot {x}^{-7}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fabs x) -1.0)) (t_1 (* (* t_0 t_0) t_0)))
   (*
    (* (pow (sqrt (PI)) -1.0) (exp (* x x)))
    (+
     (+ (+ t_0 (* (pow 2.0 -1.0) t_1)) (* (/ 3.0 4.0) (* (* t_1 t_0) t_0)))
     (* (/ 15.0 8.0) (pow x -7.0))))))

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{15}{8} \cdot {x}^{-7}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt (PI)))
  (+
   (- (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x)) (/ -0.75 (pow (fabs x) 5.0)))
   (* (/ 15.0 8.0) (pow x -7.0)))))

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{15}{8} \cdot {x}^{-7}\right) \]

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-exp.f64 100.0

       \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{15}{8} \cdot {x}^{-7}\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{15}{8} \cdot {x}^{-7}\right) \]
10. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{x}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt (PI)))
  (-
   (/ (+ (+ (/ 0.75 (pow x 4.0)) (/ 0.5 (* x x))) 1.0) (fabs x))
   (/ -1.875 (pow x 7.0)))))

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-exp.f64 100.0

       \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]
10. Step-by-step derivation

11. Applied rewrites 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{\color{blue}{{x}^{7}}}\right) \]
12. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (- (* x x) (log (sqrt (PI)))))
  (-
   (/ (+ (+ (/ 0.75 (pow x 4.0)) (/ 0.5 (* x x))) 1.0) (fabs x))
   (/ -1.875 (pow (fabs x) 7.0)))))

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right)} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs-rev N/A

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

   6. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{-x}\right)}^{\color{blue}{\left(-x\right)}}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{{\left(e^{-x}\right)}^{\left(-x\right)}}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

    4. *-lft-identity N/A

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

    5. lift-pow.f64 N/A

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

    6. lift-exp.f64 N/A

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

    7. pow-exp N/A

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

    8. lift-neg.f64 N/A

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

    9. lift-neg.f64 N/A

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

    10. sqr-neg-rev N/A

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

    11. lift-*.f64 N/A

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

    12. lift-sqrt.f64 N/A

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

    13. pow1/2 N/A

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

    14. metadata-eval N/A

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

    15. pow-to-exp N/A

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

    16. div-exp N/A

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

    17. lower-exp.f64 N/A

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

    18. lower--.f64 N/A

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

    19. rem-log-exp N/A

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

    20. pow-to-exp N/A

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

    21. metadata-eval N/A

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

11. Applied rewrites 99.9%

    \[\leadsto \color{blue}{e^{x \cdot x - \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]
12. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + 1.875 \cdot {x}^{-7}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (pow (sqrt (PI)) -1.0) (exp (* x x)))
  (+
   (- (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x)) (/ -0.75 (pow (fabs x) 5.0)))
   (* 1.875 (pow x -7.0)))))

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \color{blue}{1.875} \cdot {x}^{-7}\right) \]

9. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \color{blue}{1.875} \cdot {x}^{-7}\right) \]

10. Final simplification 99.9%

    \[\leadsto \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) + 1.875 \cdot {x}^{-7}\right) \]
11. Add Preprocessing

Alternative 7: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\left(\frac{\frac{0.75}{x \cdot x}}{x \cdot x} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (pow (sqrt (PI)) -1.0) (exp (* x x)))
  (-
   (/ (+ (+ (/ (/ 0.75 (* x x)) (* x x)) (/ 0.5 (* x x))) 1.0) (fabs x))
   (/ -1.875 (pow (fabs x) 7.0)))))

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\left(\frac{\frac{0.75}{x \cdot x}}{x \cdot x} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

10. Final simplification 99.9%

    \[\leadsto \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\left(\frac{\frac{0.75}{x \cdot x}}{x \cdot x} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]
11. Add Preprocessing

Alternative 8: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (pow (sqrt (PI)) -1.0) (exp (* x x)))
  (- (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x)) (/ -0.75 (pow (fabs x) 5.0)))))

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)} + \frac{15}{8} \cdot {x}^{-7}\right) \]

8. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)}\right) \]

   5. associate-+r- N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right) - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right) - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)} \]

10. Applied rewrites 99.0%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)} \]

11. Final simplification 99.0%

    \[\leadsto \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
12. Add Preprocessing

Alternative 9: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs-rev N/A

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

   5. lift-*.f64 98.9

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

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

Alternative 10: 21.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left({x}^{-5} \cdot 0.75\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (pow (sqrt (PI)) -1.0) (exp (* x x))) (* (pow x -5.0) 0.75)))

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fabs.f64 17.4

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

7. Applied rewrites 17.4%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{\frac{0.75}{{x}^{4}}}{\left|x\right|}} \]
8. Step-by-step derivation

9. Applied rewrites 17.4%

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

11. Applied rewrites 17.8%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left({x}^{-5} \cdot 0.75\right)} \]

12. Final simplification 17.8%

    \[\leadsto \left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{-1} \cdot e^{x \cdot x}\right) \cdot \left({x}^{-5} \cdot 0.75\right) \]
13. Add Preprocessing

Alternative 11: 21.0% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fabs.f64 17.4

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

7. Applied rewrites 17.4%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{\frac{0.75}{{x}^{4}}}{\left|x\right|}} \]
8. Step-by-step derivation

9. Applied rewrites 17.4%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{\frac{\frac{0.75}{x \cdot x}}{x \cdot x}}{\left|\color{blue}{x}\right|} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-fabs.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. sqr-abs-rev N/A

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

    5. lift-*.f64 17.4

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

11. Applied rewrites 17.4%

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

12. Final simplification 17.4%

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

Alternative 12: 1.7% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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 99.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. lower--.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right)} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs-rev N/A

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

   6. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{-x}\right)}^{\color{blue}{\left(-x\right)}}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{{\left(e^{-x}\right)}^{\left(-x\right)}}\right) \cdot \left(\frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]

10. Taylor expanded in x around 0

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

    1. associate-*r* N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. associate-*l/ N/A

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

    5. times-frac N/A

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

    6. *-rgt-identity N/A

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

    7. associate-*r/ N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. lower-pow.f64 N/A

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

    11. associate-*r/ N/A

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

    12. *-rgt-identity N/A

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

    13. lower-/.f64 N/A

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

    14. lower-sqrt.f64 N/A

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

    15. lower-/.f64 N/A

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

    16. lower-PI.f64 N/A

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

    17. lower-fabs.f64 1.8

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

12. Applied rewrites 1.8%

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

13. Final simplification 1.8%

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

Reproduce

herbie shell --seed 2024350 
(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)))))))