Jmat.Real.erf

Average Error: 13.9 → 13.8

Time: 1.3m

Precision: 64

Math

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

↓

\[\frac{{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)}^{3} + {\left(\mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)\right)}^{3}}{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) + \left(\mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right) \cdot \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right) - \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) \cdot \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)\right)}\]

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 ((pow((((1.0 / exp(pow(fabs(x), 2.0))) * ((0.284496736 / pow(fma(0.3275911, fabs(x), 1.0), 2.0)) - (1.421413741 / pow(fma(0.3275911, fabs(x), 1.0), 3.0)))) - ((1.0 / exp(pow(fabs(x), 2.0))) * ((1.061405429 / pow(fma(0.3275911, fabs(x), 1.0), 5.0)) + (0.254829592 / fma(0.3275911, fabs(x), 1.0))))), 3.0) + pow(fma(1.453152027, ((1.0 / exp(pow(fabs(x), 2.0))) / pow(fma(0.3275911, fabs(x), 1.0), 4.0)), 1.0), 3.0)) / (((((1.0 / exp(pow(fabs(x), 2.0))) * ((0.284496736 / pow(fma(0.3275911, fabs(x), 1.0), 2.0)) - (1.421413741 / pow(fma(0.3275911, fabs(x), 1.0), 3.0)))) - ((1.0 / exp(pow(fabs(x), 2.0))) * ((1.061405429 / pow(fma(0.3275911, fabs(x), 1.0), 5.0)) + (0.254829592 / fma(0.3275911, fabs(x), 1.0))))) * (((1.0 / exp(pow(fabs(x), 2.0))) * ((0.284496736 / pow(fma(0.3275911, fabs(x), 1.0), 2.0)) - (1.421413741 / pow(fma(0.3275911, fabs(x), 1.0), 3.0)))) - ((1.0 / exp(pow(fabs(x), 2.0))) * ((1.061405429 / pow(fma(0.3275911, fabs(x), 1.0), 5.0)) + (0.254829592 / fma(0.3275911, fabs(x), 1.0)))))) + ((fma(1.453152027, ((1.0 / exp(pow(fabs(x), 2.0))) / pow(fma(0.3275911, fabs(x), 1.0), 4.0)), 1.0) * fma(1.453152027, ((1.0 / exp(pow(fabs(x), 2.0))) / pow(fma(0.3275911, fabs(x), 1.0), 4.0)), 1.0)) - ((((1.0 / exp(pow(fabs(x), 2.0))) * ((0.284496736 / pow(fma(0.3275911, fabs(x), 1.0), 2.0)) - (1.421413741 / pow(fma(0.3275911, fabs(x), 1.0), 3.0)))) - ((1.0 / exp(pow(fabs(x), 2.0))) * ((1.061405429 / pow(fma(0.3275911, fabs(x), 1.0), 5.0)) + (0.254829592 / fma(0.3275911, fabs(x), 1.0))))) * fma(1.453152027, ((1.0 / exp(pow(fabs(x), 2.0))) / pow(fma(0.3275911, fabs(x), 1.0), 4.0)), 1.0)))));
}

Error

Bits error versus x

Derivation

1. Initial program 13.9

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

2. Simplified 13.9

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

3. Taylor expanded around 0 14.6

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.42141374100000006 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right)\right) - \left(1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496735999999972 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)\]

4. Simplified 13.9

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}, \frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} + 1.42141374100000006, 0.25482959199999999 - \mathsf{fma}\left(0.284496735999999972, \frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, \frac{1 \cdot 1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right)}}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)\]

5. Taylor expanded around 0 13.9

   \[\leadsto \color{blue}{\left(1 + \left(1.45315202700000001 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + 0.284496735999999972 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - \left(1.42141374100000006 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + \left(1.0614054289999999 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}} + 0.25482959199999999 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.32759110000000002 \cdot \left|x\right| + 1\right)}\right)\right)}\]

6. Simplified 13.9

   \[\leadsto \color{blue}{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) + \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)}\]
7. Using strategy rm

8. Applied flip3-+ 13.8

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)}^{3} + {\left(\mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)\right)}^{3}}{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) + \left(\mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right) \cdot \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right) - \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) \cdot \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)\right)}}\]

9. Final simplification 13.8

   \[\leadsto \frac{{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)}^{3} + {\left(\mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)\right)}^{3}}{\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) + \left(\mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right) \cdot \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right) - \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{0.284496735999999972}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{5}} + \frac{0.25482959199999999}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right) \cdot \mathsf{fma}\left(1.45315202700000001, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}, 1\right)\right)}\]

Reproduce

herbie shell --seed 2020071 +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)))))))