Average Error: 0.0 → 0.0
Time: 6.8s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \sqrt[3]{{\left(\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}\right)}^{3}}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \sqrt[3]{{\left(\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}\right)}^{3}}
double f(double x) {
        double r111008 = x;
        double r111009 = 2.30753;
        double r111010 = 0.27061;
        double r111011 = r111008 * r111010;
        double r111012 = r111009 + r111011;
        double r111013 = 1.0;
        double r111014 = 0.99229;
        double r111015 = 0.04481;
        double r111016 = r111008 * r111015;
        double r111017 = r111014 + r111016;
        double r111018 = r111017 * r111008;
        double r111019 = r111013 + r111018;
        double r111020 = r111012 / r111019;
        double r111021 = r111008 - r111020;
        return r111021;
}
↓
double f(double x) {
        double r111022 = x;
        double r111023 = 0.27061;
        double r111024 = 2.30753;
        double r111025 = fma(r111022, r111023, r111024);
        double r111026 = 1.0;
        double r111027 = 0.04481;
        double r111028 = 0.99229;
        double r111029 = fma(r111027, r111022, r111028);
        double r111030 = 1.0;
        double r111031 = fma(r111029, r111022, r111030);
        double r111032 = r111026 / r111031;
        double r111033 = r111025 * r111032;
        double r111034 = 3.0;
        double r111035 = pow(r111033, r111034);
        double r111036 = cbrt(r111035);
        double r111037 = r111022 - r111036;
        return r111037;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube21.7

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

  6. Applied cbrt-undiv21.7

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

  7. Simplified0.0

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

  9. Applied div-inv0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* (+ 0.99229 (* x 0.04481)) x)))))