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

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 5

Speedup: 0.5×

• 28.3% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right|, \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, {\left(\left|x\right|\right)}^{5} \cdot 0.2\right)\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (fma
    (fma (* x x) 0.6666666666666666 2.0)
    (fabs x)
    (fma (pow (fabs x) 7.0) 0.047619047619047616 (* (pow (fabs x) 5.0) 0.2)))
   (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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right|, \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, {\left(\left|x\right|\right)}^{5} \cdot 0.2\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right| \]

6. Final simplification 99.8%

   \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right|, \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, {\left(\left|x\right|\right)}^{5} \cdot 0.2\right)\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right| \]
7. Add Preprocessing

Alternative 2: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;\left|\left(2 \cdot \left|x\right|\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \cdot x\right) \cdot \left(0.6666666666666666 \cdot x\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.002)
   (fabs (* (* 2.0 (fabs x)) (sqrt (pow (PI) -1.0))))
   (/ (fabs (* (* x x) (* 0.6666666666666666 x))) (sqrt (PI)))))

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2e-3

   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 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. *-inverses N/A

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

      12. lower-/.f64 N/A

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

      13. *-inverses N/A

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

      14. lower-PI.f64 99.5

          \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot 2\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 98.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 99.5%

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

   if 2e-3 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow3 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 69.5

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

   7. Applied rewrites 69.5%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. rem-sqrt-square N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 69.5%

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

   11. Applied rewrites 69.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;\left|\left(2 \cdot \left|x\right|\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(x \cdot x\right) \cdot \left(0.6666666666666666 \cdot x\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (* 2.0 (fabs 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 99.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fabs.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. *-inverses N/A

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

   12. lower-/.f64 N/A

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

   13. *-inverses N/A

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

   14. lower-PI.f64 94.6

       \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot 2\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

7. Applied rewrites 94.6%

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

9. Applied rewrites 94.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 68.0%

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

13. Final simplification 68.0%

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

Alternative 4: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.8

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

   11. lift-*.f64 N/A

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

   12. lift-fabs.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. sqr-abs N/A

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

   15. lower-*.f64 99.8

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

   16. lift-/.f64 N/A

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

   17. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{0.047619047619047616} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. sqr-abs N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-fabs.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. sqr-abs N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 67.7% accurate, 5.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fabs.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. *-inverses N/A

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

   12. lower-/.f64 N/A

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

   13. *-inverses N/A

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

   14. lower-PI.f64 94.6

       \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \left|x\right| \cdot 2\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

7. Applied rewrites 94.6%

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

9. Applied rewrites 94.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 67.6%

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

Reproduce

herbie shell --seed 2024308 
(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)))))))