Average Error: 0.1 → 0.1
Time: 24.9s
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 r78325 = x;
        double r78326 = y;
        double r78327 = log(r78326);
        double r78328 = r78325 * r78327;
        double r78329 = z;
        double r78330 = r78328 + r78329;
        double r78331 = t;
        double r78332 = r78330 + r78331;
        double r78333 = a;
        double r78334 = r78332 + r78333;
        double r78335 = b;
        double r78336 = 0.5;
        double r78337 = r78335 - r78336;
        double r78338 = c;
        double r78339 = log(r78338);
        double r78340 = r78337 * r78339;
        double r78341 = r78334 + r78340;
        double r78342 = i;
        double r78343 = r78326 * r78342;
        double r78344 = r78341 + r78343;
        return r78344;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r78345 = y;
        double r78346 = i;
        double r78347 = c;
        double r78348 = log(r78347);
        double r78349 = b;
        double r78350 = 0.5;
        double r78351 = r78349 - r78350;
        double r78352 = r78348 * r78351;
        double r78353 = a;
        double r78354 = x;
        double r78355 = log(r78345);
        double r78356 = z;
        double r78357 = fma(r78354, r78355, r78356);
        double r78358 = t;
        double r78359 = r78357 + r78358;
        double r78360 = r78353 + r78359;
        double r78361 = r78352 + r78360;
        double r78362 = fma(r78345, r78346, r78361);
        return r78362;
}

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 2020047 +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)))