Average Error: 0.1 → 0.1
Time: 20.8s
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 r347 = x;
        double r348 = y;
        double r349 = log(r348);
        double r350 = r347 * r349;
        double r351 = z;
        double r352 = r350 + r351;
        double r353 = t;
        double r354 = r352 + r353;
        double r355 = a;
        double r356 = r354 + r355;
        double r357 = b;
        double r358 = 0.5;
        double r359 = r357 - r358;
        double r360 = c;
        double r361 = log(r360);
        double r362 = r359 * r361;
        double r363 = r356 + r362;
        double r364 = i;
        double r365 = r348 * r364;
        double r366 = r363 + r365;
        return r366;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r367 = i;
        double r368 = y;
        double r369 = c;
        double r370 = log(r369);
        double r371 = b;
        double r372 = 0.5;
        double r373 = r371 - r372;
        double r374 = r370 * r373;
        double r375 = x;
        double r376 = log(r368);
        double r377 = z;
        double r378 = fma(r375, r376, r377);
        double r379 = t;
        double r380 = a;
        double r381 = r379 + r380;
        double r382 = r378 + r381;
        double r383 = r374 + r382;
        double r384 = fma(r367, r368, r383);
        return r384;
}

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 2020025 +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)))