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

Percentage Accurate: 100.0% → 100.0%

Time: 9.7s

Alternatives: 5

Speedup: 1.4×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot t\_0\\ \left(\left(\left(t\_1 \cdot t\_0\right) \cdot \frac{-1}{\left|x\right|}\right) \cdot \frac{15}{8} + \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - t\_1 \cdot \frac{3}{4}\right)\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))) (t_1 (* (/ -1.0 (* (* x x) (* x x))) t_0)))
   (*
    (+
     (* (* (* t_1 t_0) (/ -1.0 (fabs x))) (/ 15.0 8.0))
     (- (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x)) (* t_1 (/ 3.0 4.0))))
    (* (exp (* (fabs x) (fabs x))) (/ 1.0 (sqrt (PI)))))))

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. *-lft-identity N/A

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

   7. associate-*l/ N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*l/ N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. lower-/.f64 N/A

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

5. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-/.f64 N/A

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

   7. inv-pow N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. lift-/.f64 N/A

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

   11. inv-pow N/A

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

   12. pow-pow N/A

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

   13. pow-prod-down N/A

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

   14. lift-fabs.f64 N/A

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

   15. lift-fabs.f64 N/A

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

   16. sqr-abs N/A

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

   17. lift-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

7. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{3}{4} \cdot \left(\color{blue}{\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-fabs.f64 N/A

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

   13. sqr-abs N/A

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

   14. lift-*.f64 N/A

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

   15. inv-pow N/A

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

   16. pow-pow N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. pow-prod-up N/A

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

9. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{3}{4} \cdot \left(\frac{-1}{\left(\left(-x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\color{blue}{\frac{-1}{\left(\left(x \cdot x\right) \cdot -1\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

10. Final simplification 100.0%

    \[\leadsto \left(\left(\left(\left(\frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{-1}{\left|x\right|}\right) \cdot \frac{15}{8} + \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \left(\frac{-1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{3}{4}\right)\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]
11. Add Preprocessing

Alternative 2: 99.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \cdot e^{x \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ -1.0 (- (/ 0.5 (fabs x)) (fabs x))) (exp (* x x))))

C

double code(double x) {
	return (-1.0 / ((0.5 / fabs(x)) - fabs(x))) * exp((x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / ((0.5d0 / abs(x)) - abs(x))) * exp((x * x))
end function

Java

public static double code(double x) {
	return (-1.0 / ((0.5 / Math.abs(x)) - Math.abs(x))) * Math.exp((x * x));
}

Python

def code(x):
	return (-1.0 / ((0.5 / math.fabs(x)) - math.fabs(x))) * math.exp((x * x))

Julia

function code(x)
	return Float64(Float64(-1.0 / Float64(Float64(0.5 / abs(x)) - abs(x))) * exp(Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / ((0.5 / abs(x)) - abs(x))) * exp((x * x));
end

Wolfram

code[x_] := N[(N[(-1.0 / N[(N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 54.3%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\mathsf{fma}\left(0.25, {\left(\left|x\right|\right)}^{-6}, \frac{-1}{x \cdot x}\right)}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-3}, 0.5, \frac{-1}{\left|x\right|}\right)}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. unpow2 N/A

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

   9. sqr-abs N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. neg-mul-1 N/A

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

   16. unsub-neg N/A

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. pow1/2 N/A

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

   13. pow-to-exp N/A

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

   14. div-exp N/A

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

   15. lower-exp.f64 N/A

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

   16. lower--.f64 N/A

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

   17. rem-log-exp N/A

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

   18. pow-to-exp N/A

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

   19. pow1/2 N/A

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

   20. lift-sqrt.f64 N/A

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

   21. lower-log.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around inf

    \[\leadsto e^{\color{blue}{{x}^{2}}} \cdot \frac{-1}{\frac{\frac{1}{2}}{\left|x\right|} - \left|x\right|} \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto e^{\color{blue}{x \cdot x}} \cdot \frac{-1}{\frac{\frac{1}{2}}{\left|x\right|} - \left|x\right|} \]

    2. lower-*.f64 99.4

       \[\leadsto e^{\color{blue}{x \cdot x}} \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \]

12. Applied rewrites 99.4%

    \[\leadsto e^{\color{blue}{x \cdot x}} \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \]

13. Final simplification 99.4%

    \[\leadsto \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \cdot e^{x \cdot x} \]
14. Add Preprocessing

Alternative 3: 76.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), t\_0 \cdot \left(x \cdot x\right), t\_0\right) \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (PI)))))
   (*
    (fma (fma 0.5 (* x x) 1.0) (* t_0 (* x x)) t_0)
    (/ -1.0 (- (/ 0.5 (fabs x)) (fabs x))))))

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 54.3%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\mathsf{fma}\left(0.25, {\left(\left|x\right|\right)}^{-6}, \frac{-1}{x \cdot x}\right)}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-3}, 0.5, \frac{-1}{\left|x\right|}\right)}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. unpow2 N/A

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

   9. sqr-abs N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. neg-mul-1 N/A

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

   16. unsub-neg N/A

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs N/A

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

   5. lift-*.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. distribute-rgt1-in N/A

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

    6. lower-fma.f64 N/A

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

12. Applied rewrites 71.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(x \cdot x\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \]
13. Add Preprocessing

Alternative 4: 52.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (fma x x 1.0) (sqrt (/ 1.0 (PI))))
  (/ -1.0 (- (/ 0.5 (fabs x)) (fabs x)))))

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 54.3%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\mathsf{fma}\left(0.25, {\left(\left|x\right|\right)}^{-6}, \frac{-1}{x \cdot x}\right)}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-3}, 0.5, \frac{-1}{\left|x\right|}\right)}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. unpow2 N/A

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

   9. sqr-abs N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. neg-mul-1 N/A

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

   16. unsub-neg N/A

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs N/A

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

   5. lift-*.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

    1. distribute-rgt1-in N/A

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

    2. lower-*.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-fma.f64 N/A

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{-1}{\frac{\frac{1}{2}}{\left|x\right|} - \left|x\right|} \]

    5. lower-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{-1}{\frac{\frac{1}{2}}{\left|x\right|} - \left|x\right|} \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{-1}{\frac{\frac{1}{2}}{\left|x\right|} - \left|x\right|} \]

    7. lower-PI.f64 47.9

       \[\leadsto \left(\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \]

12. Applied rewrites 47.9%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \]
13. Add Preprocessing

Alternative 5: 2.3% accurate, 8.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 54.3%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\color{blue}{\frac{\mathsf{fma}\left(0.25, {\left(\left|x\right|\right)}^{-6}, \frac{-1}{x \cdot x}\right)}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-3}, 0.5, \frac{-1}{\left|x\right|}\right)}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. unpow2 N/A

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

   9. sqr-abs N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. neg-mul-1 N/A

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

   16. unsub-neg N/A

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs N/A

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

   5. lift-*.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

    1. lower-sqrt.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-PI.f64 2.4

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

12. Applied rewrites 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \frac{-1}{\frac{0.5}{\left|x\right|} - \left|x\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(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)))))))