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

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

Alternatives: 12

Speedup: N/A×

• 28.4% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \left|\left(\left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616 + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot \left|x\right|, x \cdot x, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left|x\right|\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (*
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x))
     0.047619047619047616)
    (fma
     (* (* (* x x) 0.2) (fabs x))
     (* x x)
     (* (fma (* x x) 0.6666666666666666 2.0) (fabs x))))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \left|\left(\left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616 + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot \left|x\right|, x \cdot x, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left|x\right|\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right) \cdot x, x, 2\right) \cdot \left|x\right| + \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (* (fma (* (fma (* x x) 0.2 0.6666666666666666) x) x 2.0) (fabs x))
    (*
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x))
     0.047619047619047616))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

5. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. unpow2 N/A

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

   8. sqr-abs N/A

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

   9. associate-*r* N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt-out N/A

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

   12. *-commutative N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left|x\right| \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right) \cdot x, x, 2\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right) \cdot x, x, 2\right) \cdot \left|x\right| + \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
11. Add Preprocessing

Alternative 3: 99.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616 + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right), 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (*
     (* (* (* (* (* x x) (* x x)) (fabs x)) (fabs x)) (fabs x))
     0.047619047619047616)
    (* (fma (* x x) (fma 0.2 (* x x) 0.6666666666666666) 2.0) (fabs x)))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \color{blue}{0.047619047619047616} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. pow-plus N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. sqr-abs N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. distribute-lft-in N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. associate-*r* N/A

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

    8. distribute-rgt-out N/A

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

    9. *-commutative N/A

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

    10. distribute-lft-out N/A

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

    11. lower-*.f64 N/A

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

    12. lower-fabs.f64 N/A

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616 + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right), 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
13. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right| + \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (* (fma (* 0.6666666666666666 x) x 2.0) (fabs x))
    (*
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x))
     0.047619047619047616))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 99.0

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

9. Applied rewrites 99.0%

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

10. Final simplification 99.0%

    \[\leadsto \left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right| + \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
11. Add Preprocessing

Alternative 5: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right| + \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (* (fma (* 0.6666666666666666 x) x 2.0) (fabs x))
    (*
     (* (* (* (* (* x x) (* x x)) (fabs x)) (fabs x)) (fabs x))
     0.047619047619047616))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \color{blue}{0.047619047619047616} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. pow-plus N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. sqr-abs N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-out N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. +-commutative N/A

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

    7. unpow2 N/A

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

    8. associate-*r* N/A

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

    9. lower-fma.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-fabs.f64 99.0

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

11. Applied rewrites 99.0%

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

12. Final simplification 99.0%

    \[\leadsto \left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right| + \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
13. Add Preprocessing

Alternative 6: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(2 \cdot \left|x\right| + \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (* 2.0 (fabs x))
    (*
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x))
     0.047619047619047616))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 98.4

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left|x\right|} \cdot 2 + 0.047619047619047616 \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

9. Applied rewrites 98.4%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

10. Final simplification 98.4%

    \[\leadsto \left|\left(2 \cdot \left|x\right| + \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
11. Add Preprocessing

Alternative 7: 98.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left|\left(2 \cdot \left|x\right| + \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (* 2.0 (fabs x))
    (*
     (* (* (* (* (* x x) (* x x)) (fabs x)) (fabs x)) (fabs x))
     0.047619047619047616))
   (/ 1.0 (sqrt (PI))))))

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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \color{blue}{0.047619047619047616} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. pow-plus N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. sqr-abs N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. lower-fabs.f64 98.4

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 \cdot \color{blue}{\left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

11. Applied rewrites 98.4%

    \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

12. Final simplification 98.4%

    \[\leadsto \left|\left(2 \cdot \left|x\right| + \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
13. Add Preprocessing

Alternative 8: 93.3% accurate, 3.2× speedup

Math

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

FPCore

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

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

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 91.9%

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

8. Final simplification 91.9%

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

Alternative 9: 92.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right) \cdot x, x, 2\right) \cdot \left|x\right|\right|}{\sqrt{\mathsf{PI}\left(\right)}} \end{array} \]

FPCore

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

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

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 91.9%

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

9. Applied rewrites 91.5%

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

Alternative 10: 88.9% accurate, 3.9× speedup

Math

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

FPCore

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

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

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 91.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 87.2%

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

11. Final simplification 87.2%

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

Alternative 11: 88.4% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fabs (* (fma (* 0.6666666666666666 x) x 2.0) (fabs x))) (sqrt (PI))))

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

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 91.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 87.2%

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

12. Applied rewrites 86.7%

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

Alternative 12: 67.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 \cdot \left|x\right|\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (sqrt (/ 1.0 (PI))) (* 2.0 (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

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \color{blue}{0.047619047619047616} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. pow-plus N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. sqr-abs N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-fabs.f64 N/A

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

    6. lower-sqrt.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. lower-PI.f64 65.8

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

11. Applied rewrites 65.8%

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

12. Final simplification 65.8%

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

Reproduce

herbie shell --seed 2024251 
(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)))))))