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

Average Error: 2.8 → 1.4

Time: 11.0s

Precision: binary64

\[x \geq 0.5\]

Math

\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

↓

\[\left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)\right) + \frac{3}{4} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)\right)\right)\right) + \frac{15}{8} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{\sqrt{1}}{\sqrt{\left|x\right|}} \cdot {\left({\left(\frac{\sqrt{1}}{\sqrt{\left|x\right|}}\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}\right)\right)\right)\right)\right)\right)\right)\]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 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)))))))

↓

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (pow (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 (pow (fabs x) 2.0)))))))
   (*
    (/ 15.0 8.0)
    (*
     (/ 1.0 (fabs x))
     (*
      (/ 1.0 (fabs x))
      (*
       (/ 1.0 (fabs x))
       (*
        (/ 1.0 (fabs x))
        (*
         (/ 1.0 (fabs x))
         (*
          (/ (sqrt 1.0) (sqrt (fabs x)))
          (pow
           (pow (/ (sqrt 1.0) (sqrt (fabs x))) (* (cbrt 3.0) (cbrt 3.0)))
           (cbrt 3.0))))))))))))

C

double code(double x) {
	return ((double) (((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) exp(((double) (((double) fabs(x)) * ((double) fabs(x)))))))) * ((double) (((double) (((double) ((1.0 / ((double) fabs(x))) + ((double) ((1.0 / 2.0) * ((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))))))) + ((double) ((3.0 / 4.0) * ((double) (((double) (((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))))))) + ((double) ((15.0 / 8.0) * ((double) (((double) (((double) (((double) (((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x)))))))))));
}

↓

double code(double x) {
	return ((double) (((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) pow(((double) exp(((double) fabs(x)))), ((double) fabs(x)))))) * ((double) (((double) (((double) ((1.0 / ((double) fabs(x))) + ((double) ((1.0 / 2.0) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))))))))) + ((double) ((3.0 / 4.0) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) pow(((double) fabs(x)), 2.0))))))))))))) + ((double) ((15.0 / 8.0) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((1.0 / ((double) fabs(x))) * ((double) ((((double) sqrt(1.0)) / ((double) sqrt(((double) fabs(x))))) * ((double) pow(((double) pow((((double) sqrt(1.0)) / ((double) sqrt(((double) fabs(x))))), ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))), ((double) cbrt(3.0))))))))))))))))))))));
}

Error

Bits error versus x

Derivation

1. Initial program 2.8

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
2. Using strategy rm

3. Applied add-log-exp_binary64 2.8

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

4. Applied exp-to-pow_binary64 1.4

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
5. Using strategy rm

6. Applied add-sqr-sqrt_binary64 1.4

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

7. Applied add-sqr-sqrt_binary64 1.4

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

8. Applied times-frac_binary64 1.4

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

9. Applied associate-*r*_binary64 1.4

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

10. Simplified 1.4

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

11. Taylor expanded around 0 1.4

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left({\left(\frac{\sqrt{1}}{\sqrt{\left|x\right|}}\right)}^{3} \cdot \frac{\sqrt{1}}{\sqrt{\left|x\right|}}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
12. Using strategy rm

13. Applied add-cube-cbrt_binary64 1.4

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

14. Applied pow-unpow_binary64 1.4

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

15. Final simplification 1.4

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)\right) + \frac{3}{4} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)\right)\right)\right) + \frac{15}{8} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{\sqrt{1}}{\sqrt{\left|x\right|}} \cdot {\left({\left(\frac{\sqrt{1}}{\sqrt{\left|x\right|}}\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}\right)\right)\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020210 
(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)))))))