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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 7

Speedup: 2.0×

• 29.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \left|\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(0.047619047619047616 \cdot x, x, 0.2\right)}{t\_0} \cdot \left(x \cdot x\right)\right) \cdot \left(-x\right), -x, \frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{t\_0}\right) \cdot x\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (fabs
    (*
     (fma
      (* (* (/ (fma (* 0.047619047619047616 x) x 0.2) t_0) (* x x)) (- x))
      (- x)
      (/ (fma 0.6666666666666666 (* x x) 2.0) t_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 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 69.7

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

7. Applied rewrites 69.7%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

12. Applied rewrites 99.8%

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

Alternative 2: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\mathsf{PI}\left(\right)}^{-1}}\\ \left|\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, t\_0 \cdot 2\right) \cdot x\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (PI) -1.0))))
   (fabs
    (*
     (fma (* t_0 (fma (* x x) 0.2 0.6666666666666666)) (* x x) (* t_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 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 69.7

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

7. Applied rewrites 69.7%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 92.3%

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

14. Final simplification 92.3%

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

Alternative 3: 89.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

7. Applied rewrites 89.9%

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

8. Final simplification 89.9%

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

Alternative 4: 89.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (* (sqrt (pow (PI) -1.0)) (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 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 69.7

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

7. Applied rewrites 69.7%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 89.9%

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

14. Final simplification 89.9%

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

Alternative 5: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \left|\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.047619047619047616 \cdot x, x, 0.2\right)}{t\_0} \cdot \left(x \cdot x\right), x \cdot x, \frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{t\_0}\right) \cdot x\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (fabs
    (*
     (fma
      (* (/ (fma (* 0.047619047619047616 x) x 0.2) t_0) (* x x))
      (* x x)
      (/ (fma 0.6666666666666666 (* x x) 2.0) t_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 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 69.7

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

7. Applied rewrites 69.7%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

12. Applied rewrites 99.8%

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

Alternative 6: 67.5% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 69.7

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

7. Applied rewrites 69.7%

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

9. Applied rewrites 69.3%

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

11. Applied rewrites 69.7%

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

Alternative 7: 67.1% accurate, 6.8× speedup

Math

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

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 99.5%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 69.7

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

7. Applied rewrites 69.7%

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

9. Applied rewrites 69.3%

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

11. Applied rewrites 69.3%

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

Reproduce

herbie shell --seed 2024305 
(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)))))))