Average Error: 0.1 → 0.1
Time: 27.4s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\mathsf{fma}\left(x, \log y, \log t\right) - \left(z + y\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\mathsf{fma}\left(x, \log y, \log t\right) - \left(z + y\right)
double f(double x, double y, double z, double t) {
        double r62576 = x;
        double r62577 = y;
        double r62578 = log(r62577);
        double r62579 = r62576 * r62578;
        double r62580 = r62579 - r62577;
        double r62581 = z;
        double r62582 = r62580 - r62581;
        double r62583 = t;
        double r62584 = log(r62583);
        double r62585 = r62582 + r62584;
        return r62585;
}
↓
double f(double x, double y, double z, double t) {
        double r62586 = x;
        double r62587 = y;
        double r62588 = log(r62587);
        double r62589 = t;
        double r62590 = log(r62589);
        double r62591 = fma(r62586, r62588, r62590);
        double r62592 = z;
        double r62593 = r62592 + r62587;
        double r62594 = r62591 - r62593;
        return r62594;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
  2. Using strategy rm

  3. Applied pow10.1

    \[\leadsto \left(\left(x \cdot \log y - y\right) - z\right) + \log \color{blue}{\left({t}^{1}\right)}\]

  4. Applied log-pow0.1

    \[\leadsto \left(\left(x \cdot \log y - y\right) - z\right) + \color{blue}{1 \cdot \log t}\]

  5. Applied *-un-lft-identity0.1

    \[\leadsto \color{blue}{1 \cdot \left(\left(x \cdot \log y - y\right) - z\right)} + 1 \cdot \log t\]

  6. Applied distribute-lft-out0.1

    \[\leadsto \color{blue}{1 \cdot \left(\left(\left(x \cdot \log y - y\right) - z\right) + \log t\right)}\]

  7. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\left(\mathsf{fma}\left(x, \log y, \log t\right) - \left(z + y\right)\right)}\]

  8. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \left(z + y\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))