Average Error: 13.8 → 13.8
Time: 2.5m
Precision: 64
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[{e}^{\left(\log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{e^{\left|x\right| \cdot \left|x\right|} + \left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(1 + \frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}\right)\right)\right)}\]
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|}
↓
{e}^{\left(\log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{e^{\left|x\right| \cdot \left|x\right|} + \left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(1 + \frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}\right)\right)\right)}
double f(double x) {
        double r7796252 = 1.0;
        double r7796253 = 0.3275911;
        double r7796254 = x;
        double r7796255 = fabs(r7796254);
        double r7796256 = r7796253 * r7796255;
        double r7796257 = r7796252 + r7796256;
        double r7796258 = r7796252 / r7796257;
        double r7796259 = 0.254829592;
        double r7796260 = -0.284496736;
        double r7796261 = 1.421413741;
        double r7796262 = -1.453152027;
        double r7796263 = 1.061405429;
        double r7796264 = r7796258 * r7796263;
        double r7796265 = r7796262 + r7796264;
        double r7796266 = r7796258 * r7796265;
        double r7796267 = r7796261 + r7796266;
        double r7796268 = r7796258 * r7796267;
        double r7796269 = r7796260 + r7796268;
        double r7796270 = r7796258 * r7796269;
        double r7796271 = r7796259 + r7796270;
        double r7796272 = r7796258 * r7796271;
        double r7796273 = r7796255 * r7796255;
        double r7796274 = -r7796273;
        double r7796275 = exp(r7796274);
        double r7796276 = r7796272 * r7796275;
        double r7796277 = r7796252 - r7796276;
        return r7796277;
}
↓
double f(double x) {
        double r7796278 = exp(1.0);
        double r7796279 = 0.284496736;
        double r7796280 = x;
        double r7796281 = fabs(r7796280);
        double r7796282 = r7796281 * r7796281;
        double r7796283 = exp(r7796282);
        double r7796284 = r7796279 / r7796283;
        double r7796285 = 0.3275911;
        double r7796286 = r7796281 * r7796285;
        double r7796287 = 1.0;
        double r7796288 = r7796286 + r7796287;
        double r7796289 = r7796288 * r7796288;
        double r7796290 = r7796284 / r7796289;
        double r7796291 = 1.421413741;
        double r7796292 = r7796291 / r7796288;
        double r7796293 = r7796292 / r7796288;
        double r7796294 = r7796293 / r7796288;
        double r7796295 = r7796294 / r7796283;
        double r7796296 = 0.254829592;
        double r7796297 = r7796286 * r7796283;
        double r7796298 = r7796283 + r7796297;
        double r7796299 = r7796296 / r7796298;
        double r7796300 = r7796295 + r7796299;
        double r7796301 = r7796290 - r7796300;
        double r7796302 = 1.061405429;
        double r7796303 = 5.0;
        double r7796304 = pow(r7796288, r7796303);
        double r7796305 = r7796304 * r7796283;
        double r7796306 = r7796302 / r7796305;
        double r7796307 = r7796301 - r7796306;
        double r7796308 = 1.453152027;
        double r7796309 = r7796308 / r7796283;
        double r7796310 = r7796309 / r7796289;
        double r7796311 = r7796310 / r7796289;
        double r7796312 = r7796287 + r7796311;
        double r7796313 = r7796307 + r7796312;
        double r7796314 = log(r7796313);
        double r7796315 = pow(r7796278, r7796314);
        return r7796315;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.8

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

  2. Simplified13.8

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

  4. Applied add-log-exp13.8

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

  5. Applied add-log-exp13.8

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

  6. Applied diff-log14.5

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

  7. Simplified13.8

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

  8. Taylor expanded around 0 15.1

    \[\leadsto \color{blue}{\left(0.284496736 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2} \cdot e^{{\left(\left|x\right|\right)}^{2}}} + \left(1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4} \cdot e^{{\left(\left|x\right|\right)}^{2}}} + 1\right)\right) - \left(0.254829592 \cdot \frac{1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}} + \left(1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3} \cdot e^{{\left(\left|x\right|\right)}^{2}}} + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5} \cdot e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}\]

  9. Simplified13.8

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{\left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} + 1\right)}\]
  10. Using strategy rm

  11. Applied add-exp-log13.8

    \[\leadsto \color{blue}{e^{\log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{\left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} + 1\right)\right)}}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity13.8

    \[\leadsto e^{\color{blue}{1 \cdot \log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{\left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} + 1\right)\right)}}\]

  14. Applied exp-prod13.8

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{\left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} + 1\right)\right)\right)}}\]

  15. Simplified13.8

    \[\leadsto {\color{blue}{e}}^{\left(\log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{\left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} + 1\right)\right)\right)}\]

  16. Final simplification13.8

    \[\leadsto {e}^{\left(\log \left(\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(\frac{\frac{\frac{\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}}{e^{\left|x\right| \cdot \left|x\right|}} + \frac{0.254829592}{e^{\left|x\right| \cdot \left|x\right|} + \left(\left|x\right| \cdot 0.3275911\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}\right)\right) - \frac{1.061405429}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{5} \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(1 + \frac{\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)}\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))