Average Error: 0.1 → 0.1
Time: 17.6s
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 r100234 = x;
        double r100235 = y;
        double r100236 = log(r100235);
        double r100237 = r100234 * r100236;
        double r100238 = z;
        double r100239 = r100237 + r100238;
        double r100240 = t;
        double r100241 = r100239 + r100240;
        double r100242 = a;
        double r100243 = r100241 + r100242;
        double r100244 = b;
        double r100245 = 0.5;
        double r100246 = r100244 - r100245;
        double r100247 = c;
        double r100248 = log(r100247);
        double r100249 = r100246 * r100248;
        double r100250 = r100243 + r100249;
        double r100251 = i;
        double r100252 = r100235 * r100251;
        double r100253 = r100250 + r100252;
        return r100253;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r100254 = i;
        double r100255 = y;
        double r100256 = c;
        double r100257 = log(r100256);
        double r100258 = b;
        double r100259 = 0.5;
        double r100260 = r100258 - r100259;
        double r100261 = r100257 * r100260;
        double r100262 = x;
        double r100263 = log(r100255);
        double r100264 = z;
        double r100265 = fma(r100262, r100263, r100264);
        double r100266 = t;
        double r100267 = a;
        double r100268 = r100266 + r100267;
        double r100269 = r100265 + r100268;
        double r100270 = r100261 + r100269;
        double r100271 = fma(r100254, r100255, r100270);
        return r100271;
}

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 2020033 +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)))