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

Average Error: 2.8 → 1.2

Time: 4.9min

Precision: binary64

Cost: 149696

\[x \geq 0.5\]

Math

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

↓

\[\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{{x}^{1.5}}\right)}^{\left(\sqrt{x}\right)}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

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
 (*
  (sqrt (/ 1.0 PI))
  (+
   (+
    (/ (pow (exp x) x) (fabs x))
    (* 0.5 (/ (pow (exp (pow x 1.5)) (sqrt x)) (pow (fabs x) 3.0))))
   (+
    (*
     0.75
     (/
      (* (pow (* (cbrt (exp x)) (cbrt (exp x))) x) (pow (cbrt (exp x)) x))
      (pow (fabs x) 5.0)))
    (* 1.875 (/ (exp (* x x)) (pow (fabs x) 7.0)))))))

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 sqrt(1.0 / ((double) M_PI)) * (((pow(exp(x), x) / fabs(x)) + (0.5 * (pow(exp(pow(x, 1.5)), sqrt(x)) / pow(fabs(x), 3.0)))) + ((0.75 * ((pow((cbrt(exp(x)) * cbrt(exp(x))), x) * pow(cbrt(exp(x)), x)) / pow(fabs(x), 5.0))) + (1.875 * (exp(x * x) / pow(fabs(x), 7.0)))));
}

Error

Bits error versus x

Alternatives

Alternative 1
Error	1.2
Cost	124032

\[\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}} + 0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{\left(x + x\right)}}{{\left(\left|x\right|\right)}^{5}}\right)\right)\]

Alternative 2
Error	1.2
Cost	111296

\[\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}} + 0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}}\right)}^{x} \cdot e^{x \cdot \left(x \cdot 0.6666666666666666\right)}}{{\left(\left|x\right|\right)}^{5}}\right)\right)\]

Alternative 3
Error	1.2
Cost	98176

\[\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{e}^{\left(x \cdot x\right)}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

Alternative 4
Error	1.2
Cost	98112

\[\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{7}} + 0.75 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{5}}\right)\right)\]

Alternative 5
Error	1.2
Cost	98112

\[\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{7}} + 0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}}\right)\right)\]

Alternative 6
Error	1.3
Cost	33600

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

Alternative 7
Error	1.3
Cost	33600

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

Alternative 8
Error	2.7
Cost	27264

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

Alternative 9
Error	2.7
Cost	27264

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

Alternative 10
Error	2.7
Cost	27264

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

Alternative 11
Error	43.5
Cost	26880

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

Alternative 12
Error	48.2
Cost	26112

\[\sqrt{\frac{1}{\pi}} \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}\]

Alternative 13
Error	55.7
Cost	20992

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

Alternative 14
Error	56.4
Cost	20736

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

Alternative 15
Error	56.7
Cost	64

\[1\]

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

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\frac{e^{{x}^{2}}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}} + \left(1.875 \cdot \left(\frac{e^{{x}^{2}}}{{x}^{6} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.75 \cdot \left(\frac{e^{{x}^{2}}}{{x}^{4} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\]

4. Simplified 1.2

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)}\]
5. Using strategy rm

6. Applied add-cube-cbrt_binary64_2159 1.2

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}\right)}}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

7. Applied unpow-prod-down_binary64_2203 1.2

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{\color{blue}{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]
8. Using strategy rm

9. Applied pow-to-exp_binary64_2193 1.2

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{\color{blue}{e^{\log \left(e^{x}\right) \cdot x}}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

10. Simplified 1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{\color{blue}{x \cdot x}}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]
11. Using strategy rm

12. Applied add-sqr-sqrt_binary64_2146 1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

13. Applied pow-unpow_binary64_2201 1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{\color{blue}{{\left({\left(e^{x}\right)}^{\left(\sqrt{x}\right)}\right)}^{\left(\sqrt{x}\right)}}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

14. Simplified 1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\color{blue}{\left(e^{{x}^{1.5}}\right)}}^{\left(\sqrt{x}\right)}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

15. Simplified 1.2

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{{x}^{1.5}}\right)}^{\left(\sqrt{x}\right)}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)}\]

16. Final simplification 1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{{x}^{1.5}}\right)}^{\left(\sqrt{x}\right)}}{{\left(\left|x\right|\right)}^{3}}\right) + \left(0.75 \cdot \frac{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{x}}{{\left(\left|x\right|\right)}^{5}} + 1.875 \cdot \frac{e^{x \cdot x}}{{\left(\left|x\right|\right)}^{7}}\right)\right)\]

Reproduce

herbie shell --seed 2021014 
(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)))))))