Average Error: 0.1 → 0.1
Time: 11.6s
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 r87994 = x;
        double r87995 = y;
        double r87996 = log(r87995);
        double r87997 = r87994 * r87996;
        double r87998 = z;
        double r87999 = r87997 + r87998;
        double r88000 = t;
        double r88001 = r87999 + r88000;
        double r88002 = a;
        double r88003 = r88001 + r88002;
        double r88004 = b;
        double r88005 = 0.5;
        double r88006 = r88004 - r88005;
        double r88007 = c;
        double r88008 = log(r88007);
        double r88009 = r88006 * r88008;
        double r88010 = r88003 + r88009;
        double r88011 = i;
        double r88012 = r87995 * r88011;
        double r88013 = r88010 + r88012;
        return r88013;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r88014 = i;
        double r88015 = y;
        double r88016 = c;
        double r88017 = log(r88016);
        double r88018 = b;
        double r88019 = 0.5;
        double r88020 = r88018 - r88019;
        double r88021 = r88017 * r88020;
        double r88022 = x;
        double r88023 = log(r88015);
        double r88024 = z;
        double r88025 = fma(r88022, r88023, r88024);
        double r88026 = t;
        double r88027 = a;
        double r88028 = r88026 + r88027;
        double r88029 = r88025 + r88028;
        double r88030 = r88021 + r88029;
        double r88031 = fma(r88014, r88015, r88030);
        return r88031;
}

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 2020020 +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)))