Average Error: 13.7 → 13.7
Time: 7.9s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\mathsf{fma}\left(\frac{\frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, \frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\mathsf{fma}\left(1.42141374100000006, \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}, 0.25482959199999999\right) - \mathsf{fma}\left(0.284496735999999972, \frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, \frac{1}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} \cdot \frac{1.45315202700000001}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right), 1\right)\]
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{\frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, \frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\mathsf{fma}\left(1.42141374100000006, \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}, 0.25482959199999999\right) - \mathsf{fma}\left(0.284496735999999972, \frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, \frac{1}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{2}} \cdot \frac{1.45315202700000001}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right), 1\right)
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 fma(((-1.0 / fma(fabs(x), 0.3275911, 1.0)) / exp(pow(fabs(x), 2.0))), ((1.061405429 / pow(fma(0.3275911, fabs(x), 1.0), 4.0)) + (fma(1.421413741, (1.0 / pow(((0.3275911 * fabs(x)) + 1.0), 2.0)), 0.254829592) - fma(0.284496736, (1.0 / fma(0.3275911, fabs(x), 1.0)), ((1.0 / pow(fma(0.3275911, fabs(x), 1.0), 2.0)) * (1.453152027 / fma(0.3275911, fabs(x), 1.0)))))), 1.0);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.7

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

  2. Simplified13.7

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

  3. Taylor expanded around 0 14.5

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

  4. Simplified13.7

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

  6. Applied fma-udef13.7

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

  7. Simplified13.7

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

  9. Applied add-cube-cbrt13.8

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

  10. Applied unpow-prod-down13.8

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

  11. Applied times-frac13.8

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

  12. Simplified13.8

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

  13. Simplified13.7

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

  15. Applied *-un-lft-identity13.7

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

  16. Applied *-un-lft-identity13.7

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

  17. Applied distribute-lft-out13.7

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

  18. Simplified13.7

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

  19. Final simplification13.7

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

Reproduce

herbie shell --seed 2020075 +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)))))))