Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 99.5%

Time: 11.2s

Alternatives: 5

Speedup: N/A×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt (PI))) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt (PI))) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

Alternative 1: 99.5% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. sqrt-div N/A

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

   15. metadata-eval N/A

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

   16. associate-/r/ N/A

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

   17. /-rgt-identity N/A

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

   18. lift-fabs.f64 N/A

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

   19. rem-square-sqrt N/A

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

9. Applied rewrites 100.0%

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

Alternative 2: 83.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x \cdot \left(x \cdot x\right), x\right), 1\right)}{\left|x \cdot \sqrt{\mathsf{PI}\left(\right)}\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma x (fma (fma x (* x 0.16666666666666666) 0.5) (* x (* x x)) x) 1.0)
  (fabs (* x (sqrt (PI))))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. sqrt-div N/A

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

   15. metadata-eval N/A

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

   16. associate-/r/ N/A

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

   17. /-rgt-identity N/A

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

   18. lift-fabs.f64 N/A

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

   19. rem-square-sqrt N/A

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-fma.f64 N/A

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

12. Applied rewrites 90.3%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x \cdot \left(x \cdot x\right), x\right), 1\right)}}{\left|x \cdot \sqrt{\mathsf{PI}\left(\right)}\right|} \]
13. Add Preprocessing

Alternative 3: 75.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5, 1\right), 1\right)}{\left|x \cdot \sqrt{\mathsf{PI}\left(\right)}\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (* x x) (fma (* x x) 0.5 1.0) 1.0) (fabs (* x (sqrt (PI))))))

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 79.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, 1\right), 1\right) \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]

10. Applied rewrites 79.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.5, 1\right), 1\right)} \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. frac-2neg N/A

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

    7. metadata-eval N/A

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

    8. metadata-eval N/A

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

    9. frac-2neg N/A

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

    10. lift-/.f64 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. lift-fabs.f64 N/A

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

    13. clear-num N/A

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

    14. un-div-inv N/A

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

12. Applied rewrites 79.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.5, 1\right), 1\right)}{\left|x \cdot \sqrt{\mathsf{PI}\left(\right)}\right|}} \]
13. Add Preprocessing

Alternative 4: 51.5% accurate, 13.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 54.6

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

10. Applied rewrites 54.6%

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

    1. lift-fma.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. frac-2neg N/A

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

    7. lift-/.f64 N/A

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

    8. lift-sqrt.f64 N/A

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

    9. lift-fabs.f64 N/A

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

    10. div-inv N/A

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

    11. div-inv N/A

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

    12. clear-num N/A

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

    13. un-div-inv N/A

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

    14. lower-/.f64 N/A

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

    15. lift-sqrt.f64 N/A

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

    16. lift-/.f64 N/A

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

    17. sqrt-div N/A

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

    18. metadata-eval N/A

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

    19. lift-sqrt.f64 N/A

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

12. Applied rewrites 54.6%

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

Alternative 5: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-fabs.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-fabs.f64 2.2

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

10. Applied rewrites 2.2%

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

    1. lift-PI.f64 N/A

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

    2. frac-2neg N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. frac-2neg N/A

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

    6. lift-/.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. lift-fabs.f64 N/A

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

    9. clear-num N/A

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

    10. lower-/.f64 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. sqrt-div N/A

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

    14. metadata-eval N/A

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

    15. lift-sqrt.f64 N/A

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

    16. associate-/r/ N/A

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

    17. /-rgt-identity N/A

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

    18. lift-fabs.f64 N/A

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

    19. rem-square-sqrt N/A

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

    20. sqrt-prod N/A

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

    21. rem-sqrt-square N/A

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

12. Applied rewrites 2.2%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt (PI))) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))