Average Error: 0.1 → 0.1
Time: 11.4s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)\]
\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
↓
\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r73902 = x;
        double r73903 = y;
        double r73904 = log(r73903);
        double r73905 = r73902 * r73904;
        double r73906 = z;
        double r73907 = r73905 + r73906;
        double r73908 = t;
        double r73909 = r73907 + r73908;
        double r73910 = a;
        double r73911 = r73909 + r73910;
        double r73912 = b;
        double r73913 = 0.5;
        double r73914 = r73912 - r73913;
        double r73915 = c;
        double r73916 = log(r73915);
        double r73917 = r73914 * r73916;
        double r73918 = r73911 + r73917;
        double r73919 = i;
        double r73920 = r73903 * r73919;
        double r73921 = r73918 + r73920;
        return r73921;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r73922 = i;
        double r73923 = y;
        double r73924 = c;
        double r73925 = log(r73924);
        double r73926 = b;
        double r73927 = 0.5;
        double r73928 = r73926 - r73927;
        double r73929 = r73925 * r73928;
        double r73930 = x;
        double r73931 = log(r73923);
        double r73932 = z;
        double r73933 = fma(r73930, r73931, r73932);
        double r73934 = t;
        double r73935 = a;
        double r73936 = r73934 + r73935;
        double r73937 = r73933 + r73936;
        double r73938 = r73929 + r73937;
        double r73939 = fma(r73922, r73923, r73938);
        return r73939;
}

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 0.1

    \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)}\right)\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))