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

Average Error: 2.8 → 1.2

Time: 1.8min

Precision: binary64

Cost: 2880

\[x \geq 0.5\]

Math

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

↓

\[\frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{e^{0.6666666666666666 \cdot \log \left(0.5 + \frac{0.75}{x \cdot x}\right)}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

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
 (*
  (/ (* (pow (exp x) x) (sqrt (/ 1.0 PI))) (fabs x))
  (+
   1.0
   (+
    (/ 1.875 (pow x 6.0))
    (*
     (/ (exp (* 0.6666666666666666 (log (+ 0.5 (/ 0.75 (* x x)))))) x)
     (/ (cbrt (+ 0.5 (/ 0.75 (* x x)))) x))))))

C

double code(double x) {
	return ((1.0 / sqrt((double) M_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)))));
}

↓

double code(double x) {
	return ((pow(exp(x), x) * sqrt(1.0 / ((double) M_PI))) / fabs(x)) * (1.0 + ((1.875 / pow(x, 6.0)) + ((exp(0.6666666666666666 * log(0.5 + (0.75 / (x * x)))) / x) * (cbrt(0.5 + (0.75 / (x * x))) / x))));
}

Error

Bits error versus x

Alternative 1
Accuracy	1.2
Cost	3520

\[\frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{{\left(0.5 + \frac{0.75}{x \cdot x}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}\right)}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

Alternative 2
Accuracy	1.2
Cost	3008

\[\frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{{1}^{0.3333333333333333} \cdot {\left(0.5 + \frac{0.75}{x \cdot x}\right)}^{0.6666666666666666}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

Alternative 3
Accuracy	1.3
Cost	1984

\[\frac{\frac{{\left(e^{x}\right)}^{x}}{-\sqrt{\pi}}}{\left|x\right|} \cdot \left(-1 - \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)\]

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. Simplified 2.7

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)}\]

3. Taylor expanded around inf 2.7

   \[\leadsto \frac{\color{blue}{e^{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)\]

4. Simplified 1.2

   \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)\]
5. Using strategy rm

6. Applied add-cube-cbrt_binary64_4205 1.2

   \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\color{blue}{\left(\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}} \cdot \sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}\right) \cdot \sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}}{x \cdot x}\right)\right)\]

7. Applied times-frac_binary64_4176 1.2

   \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}} \cdot \sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}}\right)\right)\]
8. Using strategy rm

9. Applied pow1/3_binary64_4252 1.2

   \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\color{blue}{{\left(0.5 + \frac{0.75}{x \cdot x}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]
10. Using strategy rm

11. Applied add-exp-log_binary64_4208 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{{\left(0.5 + \frac{0.75}{x \cdot x}\right)}^{0.3333333333333333} \cdot \color{blue}{e^{\log \left(\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}\right)}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

12. Applied pow-to-exp_binary64_4239 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\color{blue}{e^{\log \left(0.5 + \frac{0.75}{x \cdot x}\right) \cdot 0.3333333333333333}} \cdot e^{\log \left(\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}\right)}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

13. Applied prod-exp_binary64_4219 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\color{blue}{e^{\log \left(0.5 + \frac{0.75}{x \cdot x}\right) \cdot 0.3333333333333333 + \log \left(\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}\right)}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

14. Simplified 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{e^{\color{blue}{0.6666666666666666 \cdot \log \left(0.5 + \frac{0.75}{x \cdot x}\right)}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

15. Final simplification 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{e^{0.6666666666666666 \cdot \log \left(0.5 + \frac{0.75}{x \cdot x}\right)}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.75}{x \cdot x}}}{x}\right)\right)\]

Reproduce

herbie shell --seed 2020322 
(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)))))))