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

Percentage Accurate: 100.0% → 99.5%

Time: 7.8s

Alternatives: 2

Speedup: N/A×

• 99.4% 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, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (* (pow (sqrt (PI)) -1.0) (exp (* x x))) (pow x -1.0)))

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

3. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}} \cdot \frac{3}{4}}\right) \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)} \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{\left|x\right|} - \frac{\frac{-15}{8}}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{\frac{-3}{4}}{{\left(\left|x\right|\right)}^{5}}\right) \]
7. Step-by-step derivation

8. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{x} - \color{blue}{\left(\frac{-0.75}{{x}^{5}} + \frac{-1.875}{{x}^{7}}\right)}\right) \]

11. Taylor expanded in x around inf

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

13. Applied rewrites 100.0%

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

14. Final simplification 100.0%

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

Alternative 2: 1.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (pow (PI) -1.0)) (/ 0.5 (pow (fabs x) 3.0))))

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

3. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}} \cdot \frac{3}{4}}\right) \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \frac{3}{4}\right)} \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.8%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{0.5}{\color{blue}{{\left(\left|x\right|\right)}^{3}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. sqrt-prod N/A

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

   4. sqr-neg-rev N/A

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

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lift-*.f64 N/A

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

   9. lower-neg.f64 1.6

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]

   10. lift-fabs.f64 N/A

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

   11. rem-sqrt-square-rev N/A

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

   12. pow2 N/A

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

   13. sqrt-pow1 N/A

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

   14. metadata-eval N/A

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

   15. unpow1 1.6

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{-\color{blue}{x} \cdot \left|x\right|}\right) \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]

   16. lift-fabs.f64 N/A

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

   17. rem-sqrt-square-rev N/A

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

   18. pow2 N/A

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

   19. sqrt-pow1 N/A

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

   20. metadata-eval N/A

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

   21. unpow1 1.6

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{-x \cdot \color{blue}{x}}\right) \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]

10. Applied rewrites 1.6%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{-x \cdot x}}\right) \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]

11. Taylor expanded in x around 0

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

    1. lower-sqrt.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-PI.f64 1.8

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]

13. Applied rewrites 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]

14. Final simplification 1.8%

    \[\leadsto \sqrt{{\mathsf{PI}\left(\right)}^{-1}} \cdot \frac{0.5}{{\left(\left|x\right|\right)}^{3}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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)))))))