Average Error: 13.8 → 13.8
Time: 10.7s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\mathsf{fma}\left(-\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\frac{\frac{1}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, 1\right)\]
1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}
↓
\mathsf{fma}\left(-\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\frac{\frac{1}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, 1\right)
double f(double x) {
        double r156395 = 1.0;
        double r156396 = 0.3275911;
        double r156397 = x;
        double r156398 = fabs(r156397);
        double r156399 = r156396 * r156398;
        double r156400 = r156395 + r156399;
        double r156401 = r156395 / r156400;
        double r156402 = 0.254829592;
        double r156403 = -0.284496736;
        double r156404 = 1.421413741;
        double r156405 = -1.453152027;
        double r156406 = 1.061405429;
        double r156407 = r156401 * r156406;
        double r156408 = r156405 + r156407;
        double r156409 = r156401 * r156408;
        double r156410 = r156404 + r156409;
        double r156411 = r156401 * r156410;
        double r156412 = r156403 + r156411;
        double r156413 = r156401 * r156412;
        double r156414 = r156402 + r156413;
        double r156415 = r156401 * r156414;
        double r156416 = r156398 * r156398;
        double r156417 = -r156416;
        double r156418 = exp(r156417);
        double r156419 = r156415 * r156418;
        double r156420 = r156395 - r156419;
        return r156420;
}
↓
double f(double x) {
        double r156421 = 1.0;
        double r156422 = x;
        double r156423 = fabs(r156422);
        double r156424 = 0.3275911;
        double r156425 = fma(r156423, r156424, r156421);
        double r156426 = r156421 / r156425;
        double r156427 = -r156426;
        double r156428 = sqrt(r156425);
        double r156429 = r156421 / r156428;
        double r156430 = r156429 / r156428;
        double r156431 = 1.061405429;
        double r156432 = -1.453152027;
        double r156433 = fma(r156430, r156431, r156432);
        double r156434 = 1.421413741;
        double r156435 = fma(r156426, r156433, r156434);
        double r156436 = -0.284496736;
        double r156437 = fma(r156435, r156426, r156436);
        double r156438 = 0.254829592;
        double r156439 = fma(r156426, r156437, r156438);
        double r156440 = r156423 * r156423;
        double r156441 = exp(r156440);
        double r156442 = r156439 / r156441;
        double r156443 = fma(r156427, r156442, r156421);
        return r156443;
}

Error

Bits error versus x

Derivation

  1. Initial program 13.8

    \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  2. Simplified13.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, 1\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt13.8

    \[\leadsto \mathsf{fma}\left(-\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, 1\right)\]

  5. Applied associate-/r*13.8

    \[\leadsto \mathsf{fma}\left(-\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, 1\right)\]

  6. Final simplification13.8

    \[\leadsto \mathsf{fma}\left(-\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\frac{\frac{1}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, 1\right)\]

Reproduce

herbie shell --seed 2020045 +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)))))))