Average Error: 2.8 → 1.2
Time: 60.0s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot {x}^{6}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right)\]
\left(\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(0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot {x}^{6}}\right) + \sqrt{\frac{1}{\pi}} \cdot \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)
(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))
   (+
    (* 0.75 (/ (pow (exp x) x) (* (pow x 4.0) (fabs x))))
    (* 1.875 (/ (pow (exp x) x) (* (fabs x) (pow x 6.0))))))
  (*
   (sqrt (/ 1.0 PI))
   (+
    (/ (pow (exp x) x) (fabs x))
    (* 0.5 (/ (pow (exp x) x) (pow (fabs x) 3.0)))))))
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)) * ((0.75 * (pow(exp(x), x) / (pow(x, 4.0) * fabs(x)))) + (1.875 * (pow(exp(x), x) / (fabs(x) * pow(x, 6.0)))))) + (sqrt(1.0 / ((double) M_PI)) * ((pow(exp(x), x) / fabs(x)) + (0.5 * (pow(exp(x), x) / pow(fabs(x), 3.0)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified2.7

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

  4. Applied add-log-exp_binary64_28452.7

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

  5. Applied exp-to-pow_binary64_28681.3

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

  6. Taylor expanded around inf 1.2

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

  7. Simplified1.2

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{6} \cdot \left|x\right|}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right)}\]

  8. Final simplification1.2

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{{\left(e^{x}\right)}^{x}}{{x}^{4} \cdot \left|x\right|} + 1.875 \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot {x}^{6}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|} + 0.5 \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\left|x\right|\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2021023 
(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)))))))