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

Percentage Accurate: 99.8% → 99.8%

Time: 15.5s

Alternatives: 13

Speedup: 2.7×

• 28.2% 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, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt (PI)))
   (fma
    x
    (fma 0.6666666666666666 (* x x) 2.0)
    (fma (pow x 7.0) 0.047619047619047616 (* 0.2 (pow x 5.0)))))))

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]
6. Step-by-step derivation

7. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

Alternative 4: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.4%

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

Alternative 5: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

10. Applied rewrites 99.4%

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

Alternative 6: 98.8% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 98.7%

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

Alternative 7: 93.3% accurate, 3.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 94.9%

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

Alternative 8: 92.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 88.7%

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

10. Applied rewrites 88.3%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 94.4%

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

Alternative 9: 68.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|\left(x + x\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (fabs (* (+ x x) (sqrt (/ 1.0 (PI)))))
   (/ (fabs (* (* (* x x) 0.6666666666666666) x)) (sqrt (PI)))))

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.0%

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

   9. Applied rewrites 65.0%

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

   if 1.75 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.7%

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

   10. Applied rewrites 88.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 29.1%

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

Alternative 10: 88.8% accurate, 4.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 88.7%

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

Alternative 11: 88.4% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 88.7%

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

10. Applied rewrites 88.3%

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

Alternative 12: 68.1% accurate, 5.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 65.0%

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

9. Applied rewrites 65.0%

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

Alternative 13: 67.6% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)}\right| \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 88.7%

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

10. Applied rewrites 88.3%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2025020 
(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)))))))