Average Error: 0.1 → 0.1
Time: 10.6s
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 r77180 = x;
        double r77181 = y;
        double r77182 = log(r77181);
        double r77183 = r77180 * r77182;
        double r77184 = z;
        double r77185 = r77183 + r77184;
        double r77186 = t;
        double r77187 = r77185 + r77186;
        double r77188 = a;
        double r77189 = r77187 + r77188;
        double r77190 = b;
        double r77191 = 0.5;
        double r77192 = r77190 - r77191;
        double r77193 = c;
        double r77194 = log(r77193);
        double r77195 = r77192 * r77194;
        double r77196 = r77189 + r77195;
        double r77197 = i;
        double r77198 = r77181 * r77197;
        double r77199 = r77196 + r77198;
        return r77199;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r77200 = i;
        double r77201 = y;
        double r77202 = c;
        double r77203 = log(r77202);
        double r77204 = b;
        double r77205 = 0.5;
        double r77206 = r77204 - r77205;
        double r77207 = r77203 * r77206;
        double r77208 = x;
        double r77209 = log(r77201);
        double r77210 = z;
        double r77211 = fma(r77208, r77209, r77210);
        double r77212 = t;
        double r77213 = a;
        double r77214 = r77212 + r77213;
        double r77215 = r77211 + r77214;
        double r77216 = r77207 + r77215;
        double r77217 = fma(r77200, r77201, r77216);
        return r77217;
}

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 2020062 +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)))