Average Error: 0.1 → 0.1
Time: 16.1s
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 r80681 = x;
        double r80682 = y;
        double r80683 = log(r80682);
        double r80684 = r80681 * r80683;
        double r80685 = z;
        double r80686 = r80684 + r80685;
        double r80687 = t;
        double r80688 = r80686 + r80687;
        double r80689 = a;
        double r80690 = r80688 + r80689;
        double r80691 = b;
        double r80692 = 0.5;
        double r80693 = r80691 - r80692;
        double r80694 = c;
        double r80695 = log(r80694);
        double r80696 = r80693 * r80695;
        double r80697 = r80690 + r80696;
        double r80698 = i;
        double r80699 = r80682 * r80698;
        double r80700 = r80697 + r80699;
        return r80700;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r80701 = y;
        double r80702 = i;
        double r80703 = c;
        double r80704 = log(r80703);
        double r80705 = b;
        double r80706 = 0.5;
        double r80707 = r80705 - r80706;
        double r80708 = r80704 * r80707;
        double r80709 = a;
        double r80710 = x;
        double r80711 = log(r80701);
        double r80712 = z;
        double r80713 = fma(r80710, r80711, r80712);
        double r80714 = t;
        double r80715 = r80713 + r80714;
        double r80716 = r80709 + r80715;
        double r80717 = r80708 + r80716;
        double r80718 = fma(r80701, r80702, r80717);
        return r80718;
}

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)))