Jmat.Real.erf

Average Error: 13.9 → 13.9

Time: 10.1s

Precision: binary64

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\frac{{1}^{3} - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}\]

FPCore

(FPCore (x)
 :precision binary64
 (-
  1.0
  (*
   (*
    (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
    (+
     0.254829592
     (*
      (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
      (+
       -0.284496736
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
        (+
         1.421413741
         (*
          (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
          (+
           -1.453152027
           (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429)))))))))
   (exp (- (* (fabs x) (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (/
  (-
   (pow 1.0 3.0)
   (pow
    (*
     (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
     (*
      (+
       0.254829592
       (+
        (/ 1.421413741 (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0))
        (-
         (/ 1.061405429 (pow (+ 1.0 (* 0.3275911 (fabs x))) 4.0))
         (+
          (/ 1.453152027 (pow (+ 1.0 (* 0.3275911 (fabs x))) 3.0))
          (/ 0.284496736 (+ 1.0 (* 0.3275911 (fabs x))))))))
      (exp (- (pow (fabs x) 2.0)))))
    3.0))
  (+
   (* 1.0 1.0)
   (*
    (*
     (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
     (*
      (+
       0.254829592
       (+
        (/ 1.421413741 (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0))
        (-
         (/ 1.061405429 (pow (+ 1.0 (* 0.3275911 (fabs x))) 4.0))
         (+
          (/ 1.453152027 (pow (+ 1.0 (* 0.3275911 (fabs x))) 3.0))
          (/ 0.284496736 (+ 1.0 (* 0.3275911 (fabs x))))))))
      (exp (- (pow (fabs x) 2.0)))))
    (+
     1.0
     (*
      (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
      (*
       (+
        0.254829592
        (+
         (/ 1.421413741 (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0))
         (-
          (/ 1.061405429 (pow (+ 1.0 (* 0.3275911 (fabs x))) 4.0))
          (+
           (/ 1.453152027 (pow (+ 1.0 (* 0.3275911 (fabs x))) 3.0))
           (/ 0.284496736 (+ 1.0 (* 0.3275911 (fabs x))))))))
       (exp (- (pow (fabs x) 2.0))))))))))

C

double code(double x) {
	return ((double) (1.0 - ((double) (((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (0.254829592 + ((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (-0.284496736 + ((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (1.421413741 + ((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (-1.453152027 + ((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * 1.061405429)))))))))))))))))) * ((double) exp(((double) -(((double) (((double) fabs(x)) * ((double) fabs(x))))))))))));
}

↓

double code(double x) {
	return (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (((double) (0.254829592 + ((double) ((1.421413741 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 2.0))) + ((double) ((1.061405429 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 4.0))) - ((double) ((1.453152027 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 3.0))) + (0.284496736 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))))))))))) * ((double) exp(((double) -(((double) pow(((double) fabs(x)), 2.0)))))))))), 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (((double) (0.254829592 + ((double) ((1.421413741 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 2.0))) + ((double) ((1.061405429 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 4.0))) - ((double) ((1.453152027 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 3.0))) + (0.284496736 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))))))))))) * ((double) exp(((double) -(((double) pow(((double) fabs(x)), 2.0)))))))))) * ((double) (1.0 + ((double) ((1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))) * ((double) (((double) (0.254829592 + ((double) ((1.421413741 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 2.0))) + ((double) ((1.061405429 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 4.0))) - ((double) ((1.453152027 / ((double) pow(((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))), 3.0))) + (0.284496736 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x))))))))))))))) * ((double) exp(((double) -(((double) pow(((double) fabs(x)), 2.0)))))))))))))))));
}

Error

Bits error versus x

Derivation

1. Initial program Error: 13.9 bits

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

2. Taylor expanded around 0 Error: 14.7 bits

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 0.254829592\right)\right) - \left(1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496736 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

3. Simplified Error: 13.9 bits

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
4. Using strategy rm

5. Applied associate-+r+ Error: 13.9 bits

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
6. Using strategy rm

7. Applied flip3-- Error: 13.9 bits

   \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}}\]

8. Simplified Error: 13.9 bits

   \[\leadsto \frac{\color{blue}{{1}^{3} - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}^{3}}}{1 \cdot 1 + \left(\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}\]

9. Simplified Error: 13.9 bits

   \[\leadsto \frac{{1}^{3} - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}}\]

10. Final simplification Error: 13.9 bits

    \[\leadsto \frac{{1}^{3} - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(\frac{1.061405429}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} - \left(\frac{1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))