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

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

Alternatives: 8

Speedup: 0.9×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left|\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \frac{1}{21} + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot \left|x\right|, x \cdot x, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left|x\right|\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (* (* (pow x 6.0) (fabs x)) (/ 1.0 21.0))
    (fma
     (* (* (* x x) 0.2) (fabs x))
     (* x x)
     (* (fma (* x x) 0.6666666666666666 2.0) (fabs x))))
   (/ 1.0 (sqrt (PI))))))

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. pow2 N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow-pow N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{x}^{6}} \cdot \left|x\right|\right)\right)\right| \]

7. Final simplification 99.9%

   \[\leadsto \left|\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \frac{1}{21} + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot \left|x\right|, x \cdot x, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left|x\right|\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. sqr-abs N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-fabs.f64 N/A

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

   13. sqr-abs N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. sqr-abs N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-fabs.f64 N/A

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

   13. sqr-abs N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-in N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-fabs.f64 99.6

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 98.8% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. sqr-abs N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-fabs.f64 N/A

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

   13. sqr-abs N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-fabs.f64 99.2

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

11. Applied rewrites 99.2%

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

12. Final simplification 99.2%

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

Alternative 5: 89.2% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. pow2 N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow-pow N/A

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

   15. lift-pow.f64 N/A

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

   16. lower-pow.f64 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({\left(\left|x\right|\right)}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 31.2%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({x}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-fabs.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 89.3

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

9. Applied rewrites 89.3%

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

Alternative 6: 88.7% accurate, 4.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. pow2 N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow-pow N/A

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

   15. lift-pow.f64 N/A

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

   16. lower-pow.f64 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({\left(\left|x\right|\right)}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 31.2%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({x}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 89.3

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

9. Applied rewrites 89.3%

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

    1. lift-fabs.f64 N/A

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

11. Applied rewrites 88.8%

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

Alternative 7: 68.5% accurate, 5.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. pow2 N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow-pow N/A

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

   15. lift-pow.f64 N/A

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

   16. lower-pow.f64 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({\left(\left|x\right|\right)}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 31.2%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({x}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 63.0

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

9. Applied rewrites 63.0%

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

10. Final simplification 63.0%

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

Alternative 8: 68.1% accurate, 5.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. pow2 N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow-pow N/A

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

   15. lift-pow.f64 N/A

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

   16. lower-pow.f64 99.9

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({\left(\left|x\right|\right)}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 31.2%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left(0.2 \cdot \left(x \cdot x\right)\right), x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left({x}^{1.5}\right)}^{4}} \cdot \left|x\right|\right)\right)\right| \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 63.0

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

9. Applied rewrites 63.0%

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

    1. lift-fabs.f64 N/A

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

11. Applied rewrites 62.5%

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

Reproduce

herbie shell --seed 2024270 
(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)))))))