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

Percentage Accurate: 99.8% → 99.9%

Time: 5.3s

Alternatives: 10

Speedup: 0.8×

• 28.6% 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: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) + {\left(\left|x\right|\right)}^{5} \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt (PI)))
   (+
    (+
     (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* x x) (fabs x))))
     (* (/ 1.0 5.0) (fabs (* (* (* (* x x) x) x) x))))
    (* (pow (fabs x) 5.0) (* (* x x) 0.047619047619047616))))))

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{{\left(\left|x\right|\right)}^{5} \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)}\right)\right| \]

5. Final simplification 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) + {\left(\left|x\right|\right)}^{5} \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right| \]
6. Add Preprocessing

Alternative 2: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x + 0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right)\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt (PI)))
   (+
    (+
     (* (fma (* x x) 0.6666666666666666 2.0) x)
     (* 0.2 (* (* (* x x) (* x x)) (fabs x))))
    (* (pow (fabs x) 6.0) (* (fabs x) 0.047619047619047616))))))

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{{\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)}\right)\right| \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. sqr-abs-rev N/A

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

   13. pow2 N/A

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

   14. associate-*r* N/A

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

   15. distribute-rgt-in N/A

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

   16. *-commutative N/A

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

6. Applied rewrites 77.5%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\color{blue}{\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x} + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 77.5

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

8. Applied rewrites 77.5%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. pow2 N/A

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

   10. associate-*l* N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 77.5

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x + 0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right)\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

10. Applied rewrites 77.5%

    \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x + 0.2 \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right)\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
11. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) + \frac{1}{21} \cdot \left(\left|\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt (PI)))
   (+
    (+
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0))
     (* 0.2 (fabs (* (* (* (* x x) x) x) x))))
    (* (/ 1.0 21.0) (* (fabs (* (* (* (* x x) (* x x)) x) x)) (fabs x)))))))

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-fabs.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. sqr-abs-rev N/A

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

   7. pow2 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. associate-*l* N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt (PI)))))
   (/
    (fabs
     (*
      (fma
       (fma (fma (* x x) 0.047619047619047616 0.2) (* x x) 0.6666666666666666)
       (* x x)
       2.0)
      x))
    (* t_0 t_0))))

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 50.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.3%

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

    1. lift-PI.f64 N/A

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

    2. lift-sqrt.f64 N/A

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

    3. add-sqr-sqrt N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. sqrt-prod N/A

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

    9. lower-*.f64 N/A

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

    10. lower-sqrt.f64 N/A

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

    11. lower-sqrt.f64 99.6

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

12. Applied rewrites 99.6%

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

Alternative 5: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 50.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.3%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-fma.f64 N/A

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

    7. associate-*r* N/A

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

    8. lower-fma.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lift-fma.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-fma.f64 N/A

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

    13. lift-*.f64 99.3

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

12. Applied rewrites 99.3%

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

Alternative 6: 98.7% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 50.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.3%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 98.5

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

13. Applied rewrites 98.5%

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

Alternative 7: 92.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 50.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 93.8%

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

Alternative 8: 67.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\left|x + x\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x\right|}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= x 1.75)
     (/ (fabs (+ x x)) t_0)
     (/ (fabs (* (* (* x x) 0.6666666666666666) x)) t_0))))

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.8%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 50.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 70.4

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

   10. Applied rewrites 70.4%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

          \[\leadsto \frac{\left|x + x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       3. lower-+.f64 70.4

          \[\leadsto \frac{\left|x + x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. Applied rewrites 70.4%

       \[\leadsto \frac{\left|x + x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   if 1.75 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   5. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      5. lift-*.f64 N/A

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

      6. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      7. lift-*.f64 N/A

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

      8. lower-*.f64 26.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fabs.f64 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      12. lift-fabs.f64 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      13. lift-fabs.f64 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      14. pow3 N/A

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

      15. lower-pow.f64 N/A

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

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

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

      17. sqrt-unprod N/A

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

      18. rem-square-sqrt 26.8

          \[\leadsto \frac{\left|0.6666666666666666 \cdot {x}^{3}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 26.8%

      \[\leadsto \frac{\left|\color{blue}{0.6666666666666666 \cdot {x}^{3}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. sqr-abs-rev N/A

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

      6. pow-prod-down N/A

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

      7. sqr-pow N/A

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

      8. pow3 N/A

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

      9. sqr-abs-rev N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. pow2 N/A

          \[\leadsto \frac{\left|\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      21. lift-*.f64 26.8

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

   9. Applied rewrites 26.8%

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

Alternative 9: 88.5% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 50.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.3%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 91.4%

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

Alternative 10: 67.0% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fabs (+ x x)) (sqrt (PI))))

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(0.047619047619047616 \cdot \left|x\right|, {\left(\left|x\right|\right)}^{6}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{2} \cdot \left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{x}^{2}} + \left(\frac{1}{5} \cdot \frac{{\left(\left|x\right|\right)}^{5}}{{x}^{2}} + \left(\frac{2}{3} \cdot \left|x\right| + 2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)\right) \cdot \color{blue}{{x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 50.8%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 70.4

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

10. Applied rewrites 70.4%

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

    1. lift-*.f64 N/A

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

    2. count-2-rev N/A

       \[\leadsto \frac{\left|x + x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

    3. lower-+.f64 70.4

       \[\leadsto \frac{\left|x + x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

12. Applied rewrites 70.4%

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

Reproduce

herbie shell --seed 2025037 
(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)))))))