Average Error: 13.5 → 13.5
Time: 8.2s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[{e}^{\left(\log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)\right)}\]
1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}
↓
{e}^{\left(\log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)\right)}
double f(double x) {
        double r183231 = 1.0;
        double r183232 = 0.3275911;
        double r183233 = x;
        double r183234 = fabs(r183233);
        double r183235 = r183232 * r183234;
        double r183236 = r183231 + r183235;
        double r183237 = r183231 / r183236;
        double r183238 = 0.254829592;
        double r183239 = -0.284496736;
        double r183240 = 1.421413741;
        double r183241 = -1.453152027;
        double r183242 = 1.061405429;
        double r183243 = r183237 * r183242;
        double r183244 = r183241 + r183243;
        double r183245 = r183237 * r183244;
        double r183246 = r183240 + r183245;
        double r183247 = r183237 * r183246;
        double r183248 = r183239 + r183247;
        double r183249 = r183237 * r183248;
        double r183250 = r183238 + r183249;
        double r183251 = r183237 * r183250;
        double r183252 = r183234 * r183234;
        double r183253 = -r183252;
        double r183254 = exp(r183253);
        double r183255 = r183251 * r183254;
        double r183256 = r183231 - r183255;
        return r183256;
}
↓
double f(double x) {
        double r183257 = exp(1.0);
        double r183258 = 1.0;
        double r183259 = 0.3275911;
        double r183260 = x;
        double r183261 = fabs(r183260);
        double r183262 = r183259 * r183261;
        double r183263 = r183258 + r183262;
        double r183264 = r183258 / r183263;
        double r183265 = 1.061405429;
        double r183266 = -1.453152027;
        double r183267 = fma(r183264, r183265, r183266);
        double r183268 = 1.421413741;
        double r183269 = fma(r183264, r183267, r183268);
        double r183270 = -0.284496736;
        double r183271 = fma(r183264, r183269, r183270);
        double r183272 = 0.254829592;
        double r183273 = fma(r183264, r183271, r183272);
        double r183274 = sqrt(r183273);
        double r183275 = r183274 * r183274;
        double r183276 = r183261 * r183261;
        double r183277 = exp(r183276);
        double r183278 = r183275 / r183277;
        double r183279 = -r183258;
        double r183280 = fma(r183261, r183259, r183258);
        double r183281 = r183279 / r183280;
        double r183282 = fma(r183278, r183281, r183258);
        double r183283 = log(r183282);
        double r183284 = pow(r183257, r183283);
        return r183284;
}

Error

Bits error versus x

Derivation

  1. Initial program 13.5

    \[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  2. Simplified13.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt13.5

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\]
  5. Using strategy rm

  6. Applied add-exp-log13.5

    \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}}\]
  7. Using strategy rm

  8. Applied pow113.5

    \[\leadsto e^{\log \color{blue}{\left({\left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}^{1}\right)}}\]

  9. Applied log-pow13.5

    \[\leadsto e^{\color{blue}{1 \cdot \log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}}\]

  10. Applied exp-prod13.5

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)\right)}}\]

  11. Simplified13.5

    \[\leadsto {\color{blue}{e}}^{\left(\log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)\right)}\]

  12. Final simplification13.5

    \[\leadsto {e}^{\left(\log \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 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)))))))