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

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 9

Speedup: 1.2×

• 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: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\\ \frac{e^{x \cdot x}}{t\_0 \cdot t\_0} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt (PI)))))
   (*
    (/ (exp (* x x)) (* t_0 t_0))
    (-
     (- (/ (- (/ 0.5 (* x x)) -1.0) (fabs x)) (/ -1.875 (pow (fabs x) 7.0)))
     (/ -0.75 (pow (fabs x) 5.0))))))

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

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 100.0

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-pow.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 100.0

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

9. Applied rewrites 100.0%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. rem-square-sqrt N/A

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

    5. lift-sqrt.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. sqrt-prod N/A

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

    8. lower-*.f64 N/A

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

    9. lower-sqrt.f64 N/A

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

    10. lower-sqrt.f64 100.0

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

11. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt (PI)))
  (-
   (- (/ (- (/ 0.5 (* x x)) -1.0) (fabs x)) (/ -1.875 (pow (fabs x) 7.0)))
   (/ -0.75 (pow (fabs x) 5.0)))))

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

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 100.0

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-pow.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 3: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt (PI)))
  (-
   (- (/ 1.0 (fabs x)) (/ -1.875 (pow (fabs x) 7.0)))
   (/ -0.75 (pow (fabs x) 5.0)))))

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

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 100.0

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-pow.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 99.7%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
13. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(\frac{1}{t\_1} \cdot e^{x \cdot x}\right) \cdot \left(\frac{0.5}{\left(x \cdot x\right) \cdot x} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, 1\right)}{t\_1} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.75 (pow (fabs x) 5.0))) (t_1 (sqrt (PI))))
   (if (<= x 2e+77)
     (* (* (/ 1.0 t_1) (exp (* x x))) (- (/ 0.5 (* (* x x) x)) t_0))
     (*
      (/ (fma (fma (* x x) 0.5 1.0) (* x x) 1.0) t_1)
      (-
       (- (/ (- (/ 0.5 (* x x)) -1.0) (fabs x)) (/ -1.875 (pow (fabs x) 7.0)))
       t_0)))))

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999997e77

   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

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

   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

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

   8. Applied rewrites 98.8%

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} - \frac{\color{blue}{-0.75}}{{\left(\left|x\right|\right)}^{5}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.8%

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

   if 1.99999999999999997e77 < x

   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

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 100.0

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. pow-exp N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, 1\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{0.5}{\left(x \cdot x\right) \cdot x} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, 1\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;x \leq 6 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{1}{t\_1} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{\left|x\right|}}{x \cdot x} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{t\_1} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.75 (pow (fabs x) 5.0))) (t_1 (sqrt (PI))))
   (if (<= x 6e+107)
     (* (* (/ 1.0 t_1) (exp (* x x))) (- (/ (/ 0.5 (fabs x)) (* x x)) t_0))
     (*
      (/ (fma x x 1.0) t_1)
      (-
       (- (/ (- (/ 0.5 (* x x)) -1.0) (fabs x)) (/ -1.875 (pow (fabs x) 7.0)))
       t_0)))))

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000046e107

   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

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

   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

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

   8. Applied rewrites 91.7%

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} - \frac{\color{blue}{-0.75}}{{\left(\left|x\right|\right)}^{5}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.1%

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

   if 6.00000000000000046e107 < x

   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

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 100.0

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. pow-exp N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{\left|x\right|}}{x \cdot x} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|} - \frac{-1.875}{{\left(\left|x\right|\right)}^{7}}\right) - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{\left|x\right|}}{x \cdot x} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt (PI))) (exp (* x x)))
  (- (/ (/ 0.5 (fabs x)) (* x x)) (/ -0.75 (pow (fabs x) 5.0)))))

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

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

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

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

8. Applied rewrites 33.7%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} - \frac{\color{blue}{-0.75}}{{\left(\left|x\right|\right)}^{5}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 36.4%

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

11. Final simplification 36.4%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{\frac{0.5}{\left|x\right|}}{x \cdot x} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
12. Add Preprocessing

Alternative 7: 33.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{0.5}{\left(x \cdot x\right) \cdot x} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt (PI))) (exp (* x x)))
  (- (/ 0.5 (* (* x x) x)) (/ -0.75 (pow (fabs x) 5.0)))))

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

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

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

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

8. Applied rewrites 33.7%

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} - \frac{\color{blue}{-0.75}}{{\left(\left|x\right|\right)}^{5}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 33.7%

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

11. Final simplification 33.7%

    \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{x \cdot x}\right) \cdot \left(\frac{0.5}{\left(x \cdot x\right) \cdot x} - \frac{-0.75}{{\left(\left|x\right|\right)}^{5}}\right) \]
12. Add Preprocessing

Alternative 8: 33.5% accurate, 1.9× speedup

Math

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

FPCore

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

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 100.0

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

   6. lift-exp.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. pow-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-pow.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 33.7%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{0.5}{\color{blue}{{\left(\left|x\right|\right)}^{3}}} \]
13. Add Preprocessing

Alternative 9: 1.8% accurate, 10.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (/ -0.5 (* (- x) x)) (* (sqrt (PI)) x)))

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

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 1.9%

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

10. Applied rewrites 1.9%

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

12. Applied rewrites 1.9%

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

13. Final simplification 1.9%

    \[\leadsto \frac{\frac{-0.5}{\left(-x\right) \cdot x}}{\sqrt{\mathsf{PI}\left(\right)} \cdot x} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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)))))))