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

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 11

Speedup: 0.5×

• 27.7% 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

\[\begin{array}{l} \\ \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ -1.0 (sqrt (PI)))
   (fma
    (pow x 6.0)
    (* x 0.047619047619047616)
    (fma 0.2 (pow x 5.0) (fma 0.6666666666666666 (pow x 3.0) (* 2.0 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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Final simplification 99.9%

   \[\leadsto \left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)\right| \]
6. Add Preprocessing

Alternative 2: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \frac{-1}{t\_0}\\ t_2 := \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;\left|t\_1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) - \frac{-1}{5} \cdot \left|t\_2\right|\right) - \frac{-1}{21} \cdot \left(\left|t\_2 \cdot x\right| \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left|t\_1 \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt{4 \cdot \left(x \cdot x\right)}\right|}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (/ -1.0 t_0)) (t_2 (* (* (* (* x x) x) x) x)))
   (if (<=
        (fabs
         (*
          t_1
          (-
           (-
            (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* x x) (fabs x))))
            (* (/ -1.0 5.0) (fabs t_2)))
           (* (/ -1.0 21.0) (* (fabs (* t_2 x)) (fabs x))))))
        2e-8)
     (fabs (* t_1 (* 2.0 x)))
     (/ (fabs (sqrt (* 4.0 (* x x)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 2e-8

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 2e-8 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (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%

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 7.3

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

   7. Applied rewrites 7.3%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. fabs-div N/A

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

      6. *-lft-identity N/A

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

      7. rem-sqrt-square N/A

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

   9. Applied rewrites 7.3%

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

   11. Applied rewrites 49.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right) - \frac{-1}{5} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{-1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt{4 \cdot \left(x \cdot x\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 69.9

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

7. Applied rewrites 69.9%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

12. Applied rewrites 99.9%

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

13. Final simplification 99.9%

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

Alternative 4: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 99.2% accurate, 2.8× speedup

Math

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

FPCore

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 69.9

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

7. Applied rewrites 69.9%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 99.5%

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

14. Final simplification 99.5%

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

Alternative 6: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 69.9

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

7. Applied rewrites 69.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

   6. *-lft-identity N/A

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

   7. rem-sqrt-square N/A

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

9. Applied rewrites 69.4%

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

10. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

    11. lower-fma.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 N/A

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

    14. unpow2 N/A

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

    15. lower-*.f64 N/A

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

    16. unpow2 N/A

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

    17. lower-*.f64 99.3

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

12. Applied rewrites 99.3%

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

Alternative 7: 93.7% accurate, 3.3× speedup

Math

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

FPCore

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 94.9

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

7. Applied rewrites 94.9%

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

8. Final simplification 94.9%

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

Alternative 8: 34.3% 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 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 94.9

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

7. Applied rewrites 94.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

9. Applied rewrites 94.4%

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

11. Applied rewrites 34.7%

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

Alternative 9: 89.8% accurate, 4.1× speedup

Math

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

FPCore

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 90.8

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

7. Applied rewrites 90.8%

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

8. Final simplification 90.8%

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

Alternative 10: 68.6% accurate, 5.4× speedup

Math

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

FPCore

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 69.9

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

7. Applied rewrites 69.9%

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

8. Final simplification 69.9%

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

Alternative 11: 68.2% accurate, 6.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left({x}^{6}, x \cdot 0.047619047619047616, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 2 \cdot x\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 69.9

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

7. Applied rewrites 69.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

   6. *-lft-identity N/A

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

   7. rem-sqrt-square N/A

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

9. Applied rewrites 69.4%

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

11. Applied rewrites 69.4%

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

Reproduce

herbie shell --seed 2024353 
(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)))))))