Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 12

Speedup: 0.6×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt (PI)))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt (PI)))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) + {5}^{-1} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ -1.0 (sqrt (PI)))
   (-
    (+
     (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* x x) (fabs x))))
     (* (pow 5.0 -1.0) (fabs (* (* (* (* x x) x) x) x))))
    (* (/ -1.0 21.0) (* (pow x 6.0) (fabs x)))))))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right)\right)\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

   4. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   5. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   6. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   7. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   8. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}\right) \cdot \left|x\right|\right)\right)\right| \]

   9. pow3 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{3}} \cdot \left|x\right|\right)\right)\right| \]

   10. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right)\right| \]

   11. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\color{blue}{\left({\left(\left|x\right|\right)}^{2}\right)}}^{3} \cdot \left|x\right|\right)\right)\right| \]

   12. pow-pow N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right)\right| \]

   13. lower-pow.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right)\right| \]

   14. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\color{blue}{\left(\left|x\right|\right)}}^{\left(2 \cdot 3\right)} \cdot \left|x\right|\right)\right)\right| \]

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

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{\left(2 \cdot 3\right)} \cdot \left|x\right|\right)\right)\right| \]

   16. sqrt-prod N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(2 \cdot 3\right)} \cdot \left|x\right|\right)\right)\right| \]

   17. rem-square-sqrt N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\color{blue}{x}}^{\left(2 \cdot 3\right)} \cdot \left|x\right|\right)\right)\right| \]

   18. metadata-eval 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({x}^{\color{blue}{6}} \cdot \left|x\right|\right)\right)\right| \]

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{x}^{6}} \cdot \left|x\right|\right)\right)\right| \]

5. Final simplification 99.9%

   \[\leadsto \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) + {5}^{-1} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]
6. Add Preprocessing

Alternative 2: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\\ \mathbf{if}\;\left|\frac{-1}{t\_0} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) + {5}^{-1} \cdot t\_1\right) - \frac{-1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 1:\\ \;\;\;\;\left|\frac{2}{t\_0} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \cdot x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (fabs (* (* (* (* x x) x) x) x))))
   (if (<=
        (fabs
         (*
          (/ -1.0 t_0)
          (-
           (+
            (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* x x) (fabs x))))
            (* (pow 5.0 -1.0) t_1))
           (* (/ -1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))
        1.0)
     (fabs (* (/ 2.0 t_0) x))
     (fabs (* (* (sqrt (pow (PI) -1.0)) (* (* x x) 0.6666666666666666)) x)))))

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 1

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.1%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 98.2

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

   if 1 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   7. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.4%

       \[\leadsto \left|\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x\right| \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 64.4%

       \[\leadsto \left|\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \cdot x\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) + {5}^{-1} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left(\left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 1:\\ \;\;\;\;\left|\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \cdot x\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\\ \left|\mathsf{fma}\left({x}^{4}, t\_0 \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), t\_0 \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (PI) -1.0))))
   (fabs
    (*
     (fma
      (pow x 4.0)
      (* t_0 (fma (* x x) 0.047619047619047616 0.2))
      (* t_0 (fma (* x x) 0.6666666666666666 2.0)))
     x))))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   2. lower-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

7. Applied rewrites 99.9%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

8. Final simplification 99.9%

   \[\leadsto \left|\mathsf{fma}\left({x}^{4}, \sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x\right| \]
9. Add Preprocessing

Alternative 4: 93.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\left(\frac{2}{t\_0} - \frac{-1}{t\_0} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot {x}^{5}\right) \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= x 2.2)
     (fabs
      (*
       (-
        (/ 2.0 t_0)
        (* (/ -1.0 t_0) (* (* (fma 0.2 (* x x) 0.6666666666666666) x) x)))
       x))
     (fabs
      (*
       (* (sqrt (pow (PI) -1.0)) (pow x 5.0))
       (fma (* x x) 0.047619047619047616 0.2))))))

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   7. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.5%

       \[\leadsto \left|\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot 2\right) \cdot x\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 92.5%

       \[\leadsto \left|\left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} - \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{{x}^{7} \cdot \left(\frac{1}{21} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{1}{5} \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{1}{5} \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \cdot {x}^{7}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{1}{5} \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \cdot {x}^{7}}\right| \]

   7. Applied rewrites 32.9%

      \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{0.2}{x \cdot x} - -0.047619047619047616\right)\right) \cdot {x}^{7}}\right| \]

   8. Taylor expanded in x around 0

      \[\leadsto \left|{x}^{5} \cdot \color{blue}{\left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 35.3%

       \[\leadsto \left|\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{5}\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} - \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot {x}^{5}\right) \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + {5}^{-1} \cdot \left|\left(t\_0 \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (fabs
    (*
     (/ -1.0 (sqrt (PI)))
     (-
      (+
       (+ (* 2.0 (fabs x)) (* (* x x) (* x 0.6666666666666666)))
       (* (pow 5.0 -1.0) (fabs (* (* t_0 x) x))))
      (* (/ -1.0 21.0) (* (* t_0 t_0) (fabs x))))))))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \frac{2}{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   6. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

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

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{\sqrt{x \cdot x}} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   8. sqrt-prod N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   9. rem-square-sqrt N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{x} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   10. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{\left|x\right|}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

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

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{\sqrt{x \cdot x}}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   12. sqrt-prod N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   13. rem-square-sqrt N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{x}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   14. lower-*.f64 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \color{blue}{\left(\left|x\right| \cdot \frac{2}{3}\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   15. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

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

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{\sqrt{x \cdot x}} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   17. sqrt-prod N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   18. rem-square-sqrt 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{x} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   19. lift-/.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\frac{2}{3}}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   20. metadata-eval 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{0.6666666666666666}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

4. Applied rewrites 78.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   5. sqr-abs-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   6. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   7. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   8. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   9. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   10. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   11. sqr-abs-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   12. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   13. lower-*.f64 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 78.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right)\right)\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{\left(x \cdot x\right)}^{2}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   5. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left({\color{blue}{\left(x \cdot x\right)}}^{2} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   6. unpow-prod-down N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   7. pow-sqr N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   8. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left({x}^{\color{blue}{4}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   9. lift-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{x}^{4}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

   11. lift-pow.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{{x}^{4}} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   12. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left({x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   13. pow-pow N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{{\left({x}^{2}\right)}^{2}} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   14. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left({\color{blue}{\left(x \cdot x\right)}}^{2} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   15. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left({\color{blue}{\left(x \cdot x\right)}}^{2} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   16. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   17. sqr-neg-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   18. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   19. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right) \cdot \left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   20. sqr-abs-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right)\right)\right| \]

   21. unswap-sqr N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

   22. lower-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

8. Applied rewrites 78.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

9. Final simplification 78.8%

   \[\leadsto \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + {5}^{-1} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]
10. Add Preprocessing

Alternative 6: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + {5}^{-1} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right)\right) - \frac{-1}{21} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (fabs
    (*
     (/ -1.0 (sqrt (PI)))
     (-
      (+
       (+ (* 2.0 (fabs x)) (* (* x x) (* x 0.6666666666666666)))
       (* (pow 5.0 -1.0) (* (* (* x x) (* x x)) (fabs x))))
      (* (/ -1.0 21.0) (* (* t_0 t_0) (fabs x))))))))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \frac{2}{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   6. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

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

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{\sqrt{x \cdot x}} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   8. sqrt-prod N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   9. rem-square-sqrt N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\color{blue}{x} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   10. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{\left|x\right|}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

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

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{\sqrt{x \cdot x}}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   12. sqrt-prod N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   13. rem-square-sqrt N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot \color{blue}{x}\right) \cdot \left(\left|x\right| \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   14. lower-*.f64 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \color{blue}{\left(\left|x\right| \cdot \frac{2}{3}\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   15. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

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

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{\sqrt{x \cdot x}} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   17. sqrt-prod N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   18. rem-square-sqrt 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(\color{blue}{x} \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   19. lift-/.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\frac{2}{3}}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   20. metadata-eval 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{0.6666666666666666}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

4. Applied rewrites 78.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   5. sqr-abs-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   6. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   7. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   8. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   9. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   10. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   11. sqr-abs-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   12. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   13. lower-*.f64 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 78.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right)\right)\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{\left(x \cdot x\right)}^{2}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   5. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left({\color{blue}{\left(x \cdot x\right)}}^{2} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   6. unpow-prod-down N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   7. pow-sqr N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   8. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left({x}^{\color{blue}{4}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   9. lift-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{x}^{4}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

   11. lift-pow.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{{x}^{4}} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   12. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left({x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   13. pow-pow N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{{\left({x}^{2}\right)}^{2}} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   14. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left({\color{blue}{\left(x \cdot x\right)}}^{2} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   15. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left({\color{blue}{\left(x \cdot x\right)}}^{2} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   16. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   17. sqr-neg-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   18. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   19. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right) \cdot \left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   20. sqr-abs-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right)\right)\right| \]

   21. unswap-sqr N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

   22. lower-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot x\right)\right)} \cdot \left|x\right|\right)\right)\right| \]

8. Applied rewrites 78.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right)} \cdot \left|x\right|\right)\right)\right| \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\color{blue}{\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   5. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   6. sqr-abs-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   7. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   8. associate-*l* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)} \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   9. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   10. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   11. sqr-abs-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   12. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) + \frac{1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

   13. lower-*.f64 78.8

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

10. Applied rewrites 78.8%

    \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + \frac{1}{5} \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(-x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]

11. Final simplification 78.8%

    \[\leadsto \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right)\right) + {5}^{-1} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right)\right) - \frac{-1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right)\right| \]
12. Add Preprocessing

Alternative 7: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;\left|\left(\frac{2}{t\_0} - \frac{-1}{t\_0} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7} \cdot 0.047619047619047616}{t\_0}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= x 2.65)
     (fabs
      (*
       (-
        (/ 2.0 t_0)
        (* (/ -1.0 t_0) (* (* (fma 0.2 (* x x) 0.6666666666666666) x) x)))
       x))
     (fabs (/ (* (pow x 7.0) 0.047619047619047616) t_0)))))

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999991

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   7. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.5%

       \[\leadsto \left|\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot 2\right) \cdot x\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 92.5%

       \[\leadsto \left|\left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} - \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right| \]

   if 2.64999999999999991 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right| \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left({x}^{7} \cdot \frac{1}{21}\right)} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left({x}^{7} \cdot \frac{1}{21}\right)} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-pow.f64 N/A

         \[\leadsto \left|\left(\color{blue}{{x}^{7}} \cdot \frac{1}{21}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right| \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left|\left({x}^{7} \cdot \frac{1}{21}\right) \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left({x}^{7} \cdot \frac{1}{21}\right) \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      8. lower-PI.f64 35.0

         \[\leadsto \left|\left({x}^{7} \cdot 0.047619047619047616\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

   7. Applied rewrites 35.0%

      \[\leadsto \left|\color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

   9. Applied rewrites 35.0%

      \[\leadsto \left|\frac{{x}^{7} \cdot 0.047619047619047616}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;\left|\left(\frac{2}{t\_0} - \frac{-1}{t\_0} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{t\_0}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= x 2.65)
     (fabs
      (*
       (-
        (/ 2.0 t_0)
        (* (/ -1.0 t_0) (* (* (fma 0.2 (* x x) 0.6666666666666666) x) x)))
       x))
     (fabs (* 0.047619047619047616 (/ (pow x 7.0) t_0))))))

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999991

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   7. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.5%

       \[\leadsto \left|\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot 2\right) \cdot x\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 92.5%

       \[\leadsto \left|\left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} - \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right| \]

   if 2.64999999999999991 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   5. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right| \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left({x}^{7} \cdot \frac{1}{21}\right)} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left({x}^{7} \cdot \frac{1}{21}\right)} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-pow.f64 N/A

         \[\leadsto \left|\left(\color{blue}{{x}^{7}} \cdot \frac{1}{21}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right| \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left|\left({x}^{7} \cdot \frac{1}{21}\right) \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left({x}^{7} \cdot \frac{1}{21}\right) \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

      8. lower-PI.f64 35.0

         \[\leadsto \left|\left({x}^{7} \cdot 0.047619047619047616\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

   7. Applied rewrites 35.0%

      \[\leadsto \left|\color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 35.0%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\frac{{x}^{7}}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left|\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (* (fma (* x x) 0.6666666666666666 2.0) x) (sqrt (pow (PI) -1.0)))))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-/.f64 N/A

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

7. Applied rewrites 87.8%

   \[\leadsto \left|\color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

8. Final simplification 87.8%

   \[\leadsto \left|\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right| \]
9. Add Preprocessing

Alternative 10: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left|\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (* (sqrt (pow (PI) -1.0)) (fma (* x x) 0.6666666666666666 2.0)) x)))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   2. lower-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

7. Applied rewrites 99.9%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 87.8%

    \[\leadsto \left|\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x\right| \]

11. Final simplification 87.8%

    \[\leadsto \left|\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x\right| \]
12. Add Preprocessing

Alternative 11: 93.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \left|\left(\frac{2}{t\_0} - \frac{-1}{t\_0} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (fabs
    (*
     (-
      (/ 2.0 t_0)
      (* (/ -1.0 t_0) (* (* (fma 0.2 (* x x) 0.6666666666666666) x) x)))
     x))))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right| \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

   2. lower-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{2}{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \frac{1}{5} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \cdot x}\right| \]

7. Applied rewrites 99.9%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left({x}^{4}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot x}\right| \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 92.5%

    \[\leadsto \left|\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot 2\right) \cdot x\right| \]
11. Step-by-step derivation

12. Applied rewrites 92.5%

    \[\leadsto \left|\left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} - \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right)\right) \cdot x\right| \]
13. Add Preprocessing

Alternative 12: 68.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (/ 2.0 (sqrt (PI))) x)))

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), {x}^{7} \cdot 0.047619047619047616\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 68.9

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

7. Applied rewrites 68.9%

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

9. Applied rewrites 68.9%

   \[\leadsto \color{blue}{\left|\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right|} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024344 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt (PI))) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))