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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 11

Speedup: 1.2×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{\frac{0.5}{x \cdot x} - -1}{\sqrt{x}}}{\sqrt{x}} + \frac{3}{4} \cdot \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt (PI)))
  (+
   (+
    (/ (/ (- (/ 0.5 (* x x)) -1.0) (sqrt x)) (sqrt x))
    (* (/ 3.0 4.0) (* (/ 1.0 (* (* x x) (* x x))) (/ 1.0 (fabs x)))))
   (/ 1.875 (pow (fabs x) 7.0)))))

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 100.0

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-fabs.f64 100.0

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{\frac{0.5}{x \cdot x} - -1}{\sqrt{x}}}{\sqrt{x}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{1.875}{{\color{blue}{\left(\left|x\right|\right)}}^{7}}\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{\frac{0.5}{x \cdot x} - -1}{\sqrt{x}}}{\sqrt{x}} + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \color{blue}{\frac{1.875}{{\left(\left|x\right|\right)}^{7}}}\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-fabs.f64 N/A

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

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

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

    5. sqrt-unprod N/A

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

    6. rem-square-sqrt N/A

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

    7. lift-*.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. lift-fabs.f64 N/A

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

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

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

    11. sqrt-unprod N/A

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

    12. rem-square-sqrt N/A

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

    13. associate-*r/ N/A

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

    14. frac-times N/A

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

11. Applied rewrites 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{\frac{0.5}{x \cdot x} - -1}{\sqrt{x}}}{\sqrt{x}} + \frac{3}{4} \cdot \left(\color{blue}{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\left(\left(\frac{0.75}{{x}^{4}} + 1\right) - \frac{-0.5}{x \cdot x}\right) - \frac{-1.875}{{x}^{6}}}{x} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot e^{x \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (/
    (-
     (- (+ (/ 0.75 (pow x 4.0)) 1.0) (/ -0.5 (* x x)))
     (/ -1.875 (pow x 6.0)))
    x)
   (sqrt (/ 1.0 (PI))))
  (exp (* x x))))

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot e^{x \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 100.0%

    \[\leadsto \left(\frac{\left(\left(\frac{0.75}{{x}^{4}} + 1\right) - \frac{-0.5}{x \cdot x}\right) - \frac{-1.875}{{x}^{6}}}{x} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot e^{\color{blue}{x} \cdot x} \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\\ \frac{\mathsf{fma}\left(t\_0, \frac{\frac{\frac{0.75}{x \cdot x} - -0.5}{x}}{x}, t\_0\right)}{x} \cdot e^{x \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (PI)))))
   (*
    (/ (fma t_0 (/ (/ (- (/ 0.75 (* x x)) -0.5) x) x) t_0) x)
    (exp (* x x)))))

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot e^{x \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, \frac{\frac{\frac{0.75}{x \cdot x} - -0.5}{x}}{x}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{x} \cdot e^{\color{blue}{x \cdot x}} \]
13. Add Preprocessing

Alternative 4: 99.6% accurate, 2.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot e^{x \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 99.5%

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

Alternative 5: 99.4% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (exp (* x x)) (* (sqrt (PI)) x)))

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 6: 99.2% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (pow (+ 1.0 x) x) (* (sqrt (PI)) x)))

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.9%

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

13. Final simplification 98.9%

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

Alternative 7: 75.5% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 76.9%

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

11. Final simplification 76.9%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{x} \]
12. Add Preprocessing

Alternative 8: 75.5% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 76.9%

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

11. Taylor expanded in x around inf

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{x} \]
12. Step-by-step derivation

13. Applied rewrites 76.9%

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

14. Final simplification 76.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{x} \]
15. Add Preprocessing

Alternative 9: 68.1% accurate, 10.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 76.9%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 66.8%

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

14. Final simplification 66.8%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot x \]
15. Add Preprocessing

Alternative 10: 51.9% accurate, 14.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 53.8%

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

13. Final simplification 53.8%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\mathsf{PI}\left(\right)} \cdot x} \]
14. Add Preprocessing

Alternative 11: 2.3% accurate, 17.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. unpow2 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 2.3%

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

13. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2025017 
(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)))))))