Average Error: 28.7 → 28.8
Time: 16.2s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72996 = x;
        double r72997 = y;
        double r72998 = r72996 * r72997;
        double r72999 = z;
        double r73000 = r72998 + r72999;
        double r73001 = r73000 * r72997;
        double r73002 = 27464.7644705;
        double r73003 = r73001 + r73002;
        double r73004 = r73003 * r72997;
        double r73005 = 230661.510616;
        double r73006 = r73004 + r73005;
        double r73007 = r73006 * r72997;
        double r73008 = t;
        double r73009 = r73007 + r73008;
        double r73010 = a;
        double r73011 = r72997 + r73010;
        double r73012 = r73011 * r72997;
        double r73013 = b;
        double r73014 = r73012 + r73013;
        double r73015 = r73014 * r72997;
        double r73016 = c;
        double r73017 = r73015 + r73016;
        double r73018 = r73017 * r72997;
        double r73019 = i;
        double r73020 = r73018 + r73019;
        double r73021 = r73009 / r73020;
        return r73021;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r73022 = x;
        double r73023 = y;
        double r73024 = z;
        double r73025 = fma(r73022, r73023, r73024);
        double r73026 = 27464.7644705;
        double r73027 = fma(r73025, r73023, r73026);
        double r73028 = 230661.510616;
        double r73029 = fma(r73027, r73023, r73028);
        double r73030 = t;
        double r73031 = fma(r73029, r73023, r73030);
        double r73032 = 1.0;
        double r73033 = a;
        double r73034 = r73023 + r73033;
        double r73035 = b;
        double r73036 = fma(r73034, r73023, r73035);
        double r73037 = c;
        double r73038 = fma(r73036, r73023, r73037);
        double r73039 = i;
        double r73040 = fma(r73038, r73023, r73039);
        double r73041 = r73032 / r73040;
        double r73042 = r73031 * r73041;
        return r73042;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 28.7

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  2. Simplified28.7

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]
  3. Using strategy rm

  4. Applied div-inv28.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]

  5. Final simplification28.8

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))