Average Error: 0.1 → 0.1
Time: 10.8s
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 r77194 = x;
        double r77195 = y;
        double r77196 = log(r77195);
        double r77197 = r77194 * r77196;
        double r77198 = z;
        double r77199 = r77197 + r77198;
        double r77200 = t;
        double r77201 = r77199 + r77200;
        double r77202 = a;
        double r77203 = r77201 + r77202;
        double r77204 = b;
        double r77205 = 0.5;
        double r77206 = r77204 - r77205;
        double r77207 = c;
        double r77208 = log(r77207);
        double r77209 = r77206 * r77208;
        double r77210 = r77203 + r77209;
        double r77211 = i;
        double r77212 = r77195 * r77211;
        double r77213 = r77210 + r77212;
        return r77213;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r77214 = i;
        double r77215 = y;
        double r77216 = c;
        double r77217 = log(r77216);
        double r77218 = b;
        double r77219 = 0.5;
        double r77220 = r77218 - r77219;
        double r77221 = r77217 * r77220;
        double r77222 = x;
        double r77223 = log(r77215);
        double r77224 = z;
        double r77225 = fma(r77222, r77223, r77224);
        double r77226 = t;
        double r77227 = a;
        double r77228 = r77226 + r77227;
        double r77229 = r77225 + r77228;
        double r77230 = r77221 + r77229;
        double r77231 = fma(r77214, r77215, r77230);
        return r77231;
}

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)))