Average Error: 13.9 → 13.9
Time: 11.5s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\log \left(e^{1 - \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \sqrt[3]{{\left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \sqrt{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999} \cdot \sqrt{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999}\right)\right)\right)}^{3}}\right) \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}\right)\]
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|}
↓
\log \left(e^{1 - \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \sqrt[3]{{\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 + \sqrt{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999} \cdot \sqrt{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999}\right)\right)\right)}^{3}}\right) \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}\right)
double code(double x) {
	return ((double) (1.0 - ((double) (((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (0.254829592 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (-0.284496736 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (1.421413741 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (-1.453152027 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * 1.061405429)))))))))))))))))) * ((double) exp(((double) -(((double) (((double) fabs(x)) * ((double) fabs(x))))))))))));
}
↓
double code(double x) {
	return ((double) log(((double) exp(((double) (1.0 - ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (((double) (0.254829592 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) cbrt(((double) pow(((double) (-0.284496736 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (1.421413741 + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * ((double) (-1.453152027 + ((double) (((double) sqrt(((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * 1.061405429)))) * ((double) sqrt(((double) (((double) (1.0 / ((double) (1.0 + ((double) (0.3275911 * ((double) fabs(x)))))))) * 1.061405429)))))))))))))))), 3.0)))))))) * ((double) (1.0 / ((double) exp(((double) pow(((double) fabs(x)), 2.0))))))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.9

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

  3. Applied add-log-exp13.9

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

  4. Applied add-log-exp13.9

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

  5. Applied diff-log14.6

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

  6. Simplified13.9

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

  8. Applied add-sqr-sqrt13.9

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

  10. Applied add-cbrt-cube13.9

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

  11. Simplified13.9

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

  12. Final simplification13.9

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

Reproduce

herbie shell --seed 2020121 
(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)))))))