Average Error: 0.1 → 0.1
Time: 29.3s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\mathsf{fma}\left(y, i, \log c \cdot \left(b - 0.5\right) + \left(a + \left(\mathsf{fma}\left(x, \log y, z\right) + t\right)\right)\right)\]
\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(y, i, \log c \cdot \left(b - 0.5\right) + \left(a + \left(\mathsf{fma}\left(x, \log y, z\right) + t\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r62309 = x;
        double r62310 = y;
        double r62311 = log(r62310);
        double r62312 = r62309 * r62311;
        double r62313 = z;
        double r62314 = r62312 + r62313;
        double r62315 = t;
        double r62316 = r62314 + r62315;
        double r62317 = a;
        double r62318 = r62316 + r62317;
        double r62319 = b;
        double r62320 = 0.5;
        double r62321 = r62319 - r62320;
        double r62322 = c;
        double r62323 = log(r62322);
        double r62324 = r62321 * r62323;
        double r62325 = r62318 + r62324;
        double r62326 = i;
        double r62327 = r62310 * r62326;
        double r62328 = r62325 + r62327;
        return r62328;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r62329 = y;
        double r62330 = i;
        double r62331 = c;
        double r62332 = log(r62331);
        double r62333 = b;
        double r62334 = 0.5;
        double r62335 = r62333 - r62334;
        double r62336 = r62332 * r62335;
        double r62337 = a;
        double r62338 = x;
        double r62339 = log(r62329);
        double r62340 = z;
        double r62341 = fma(r62338, r62339, r62340);
        double r62342 = t;
        double r62343 = r62341 + r62342;
        double r62344 = r62337 + r62343;
        double r62345 = r62336 + r62344;
        double r62346 = fma(r62329, r62330, r62345);
        return r62346;
}

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

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

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 +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)))