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

Percentage Accurate: 99.8% → 99.8%

Time: 14.9s

Alternatives: 11

Speedup: 0.8×

• 28.9% 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, 0.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow3 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. pow-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-fabs.f64 N/A

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

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

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

   16. sqrt-prod N/A

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

   17. rem-square-sqrt N/A

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

   18. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 76.7% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

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

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

   8. sqrt-prod N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-fabs.f64 N/A

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

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

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

   12. sqrt-prod N/A

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

   13. rem-square-sqrt N/A

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

   14. lower-*.f64 99.8

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

   15. lift-fabs.f64 N/A

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

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

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

   17. sqrt-prod N/A

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

   18. rem-square-sqrt 75.2

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

   19. lift-/.f64 N/A

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

   20. metadata-eval 75.2

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

4. Applied rewrites 75.2%

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

5. Final simplification 75.2%

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

Alternative 3: 88.8% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

7. Applied rewrites 90.0%

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

8. Final simplification 90.0%

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

Alternative 4: 67.1% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lower-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 66.9

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

7. Applied rewrites 66.9%

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

8. Final simplification 66.9%

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

Alternative 5: 99.4% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 6: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 7: 98.6% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 98.9%

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

Alternative 8: 92.8% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 93.0

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

7. Applied rewrites 93.0%

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

9. Applied rewrites 93.0%

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

11. Applied rewrites 93.0%

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

Alternative 9: 33.8% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 93.0

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

7. Applied rewrites 93.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 32.0

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

9. Applied rewrites 32.0%

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

Alternative 10: 88.3% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 89.6

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

7. Applied rewrites 89.6%

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

9. Applied rewrites 89.6%

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

Alternative 11: 33.6% accurate, 7.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 66.5

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

7. Applied rewrites 66.5%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 31.9

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

9. Applied rewrites 31.9%

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

11. Applied rewrites 31.9%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))