Average Error: 0.0 → 0.1
Time: 6.8s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \sqrt[3]{\frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}} \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}}\right)\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.707110000000000016, \sqrt[3]{\frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}} \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}}\right)
double f(double x) {
        double r143738 = 0.70711;
        double r143739 = 2.30753;
        double r143740 = x;
        double r143741 = 0.27061;
        double r143742 = r143740 * r143741;
        double r143743 = r143739 + r143742;
        double r143744 = 1.0;
        double r143745 = 0.99229;
        double r143746 = 0.04481;
        double r143747 = r143740 * r143746;
        double r143748 = r143745 + r143747;
        double r143749 = r143740 * r143748;
        double r143750 = r143744 + r143749;
        double r143751 = r143743 / r143750;
        double r143752 = r143751 - r143740;
        double r143753 = r143738 * r143752;
        return r143753;
}
↓
double f(double x) {
        double r143754 = x;
        double r143755 = -r143754;
        double r143756 = 0.70711;
        double r143757 = 0.04481;
        double r143758 = 0.99229;
        double r143759 = fma(r143757, r143754, r143758);
        double r143760 = 1.0;
        double r143761 = fma(r143754, r143759, r143760);
        double r143762 = sqrt(r143761);
        double r143763 = 0.27061;
        double r143764 = 2.30753;
        double r143765 = fma(r143763, r143754, r143764);
        double r143766 = r143762 / r143765;
        double r143767 = r143756 / r143766;
        double r143768 = r143767 / r143761;
        double r143769 = r143767 * r143767;
        double r143770 = r143769 / r143762;
        double r143771 = r143768 * r143770;
        double r143772 = cbrt(r143771);
        double r143773 = fma(r143755, r143756, r143772);
        return r143773;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}}\right)\]

  5. Simplified0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\color{blue}{\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}} \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}}\right)\]

  8. Applied associate-/r*0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \color{blue}{\frac{\frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}}{\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}}\right)\]

  9. Simplified0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{\color{blue}{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}}{\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}\right)\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}} \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}\right) \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}}}\right)\]

  12. Applied add-cbrt-cube0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{\color{blue}{\sqrt[3]{\left(\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}} \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right) \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}}}{\sqrt[3]{\left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}} \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}\right) \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}}\right)\]

  13. Applied cbrt-undiv0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \color{blue}{\sqrt[3]{\frac{\left(\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}} \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right) \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}} \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}\right) \cdot \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}}}}\right)\]

  14. Simplified0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \sqrt[3]{\color{blue}{\frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}} \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}}}\right)\]

  15. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \sqrt[3]{\frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \frac{\frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}} \cdot \frac{0.707110000000000016}{\frac{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}}\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))