Jmat.Real.erf

Average Error: 13.5 → 10.4

Time: 39.9s

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

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
 (/
  (-
   1.0
   (*
    (/
     (cbrt
      (*
       (*
        (+
         0.254829592
         (/
          (+
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
              (+ 1.0 (* (fabs x) 0.3275911))))
            (+ 1.0 (* (fabs x) 0.3275911)))
           -0.284496736)
          (+ 1.0 (* (fabs x) 0.3275911))))
        (+
         0.254829592
         (/
          (+
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
              (+ 1.0 (* (fabs x) 0.3275911))))
            (+ 1.0 (* (fabs x) 0.3275911)))
           -0.284496736)
          (+ 1.0 (* (fabs x) 0.3275911)))))
       (*
        (*
         (+
          0.254829592
          (/
           (+
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
               (+ 1.0 (* (fabs x) 0.3275911))))
             (+ 1.0 (* (fabs x) 0.3275911)))
            -0.284496736)
           (+ 1.0 (* (fabs x) 0.3275911))))
         (+
          0.254829592
          (/
           (+
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
               (+ 1.0 (* (fabs x) 0.3275911))))
             (+ 1.0 (* (fabs x) 0.3275911)))
            -0.284496736)
           (+ 1.0 (* (fabs x) 0.3275911)))))
        (*
         (+
          0.254829592
          (/
           (+
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
               (+ 1.0 (* (fabs x) 0.3275911))))
             (+ 1.0 (* (fabs x) 0.3275911)))
            -0.284496736)
           (+ 1.0 (* (fabs x) 0.3275911))))
         (+
          0.254829592
          (/
           (+
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
               (+ 1.0 (* (fabs x) 0.3275911))))
             (+ 1.0 (* (fabs x) 0.3275911)))
            -0.284496736)
           (+ 1.0 (* (fabs x) 0.3275911))))))))
     (pow (+ 1.0 (* (fabs x) 0.3275911)) 2.0))
    (exp (* (* x x) -2.0))))
  (+
   1.0
   (/
    (+
     0.254829592
     (/
      (+
       (/
        (+
         1.421413741
         (/
          (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs x) 0.3275911))))
          (+ 1.0 (* (fabs x) 0.3275911))))
        (+ 1.0 (* (fabs x) 0.3275911)))
       -0.284496736)
      (+ 1.0 (* (fabs x) 0.3275911))))
    (* (+ 1.0 (* (fabs x) 0.3275911)) (exp (* x x)))))))

C

double code(double x) {
	return 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))));
}

↓

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

Error

Bits error versus x

Derivation

1. Initial program 13.5

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

3. Applied flip--_binary64_2781 13.5

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

4. Simplified 13.5

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

5. Simplified 13.5

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

7. Applied add-cbrt-cube_binary64_2842 10.4

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

8. Final simplification 10.4

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

Reproduce

herbie shell --seed 2021098 
(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)))))))