Average Error: 0.1 → 0.1
Time: 11.1s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(-\mathsf{fma}\left(x, -\log y, y\right)\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(-\mathsf{fma}\left(x, -\log y, y\right)\right) - z\right) + \log t
double f(double x, double y, double z, double t) {
        double r121462 = x;
        double r121463 = y;
        double r121464 = log(r121463);
        double r121465 = r121462 * r121464;
        double r121466 = r121465 - r121463;
        double r121467 = z;
        double r121468 = r121466 - r121467;
        double r121469 = t;
        double r121470 = log(r121469);
        double r121471 = r121468 + r121470;
        return r121471;
}
↓
double f(double x, double y, double z, double t) {
        double r121472 = x;
        double r121473 = y;
        double r121474 = log(r121473);
        double r121475 = -r121474;
        double r121476 = fma(r121472, r121475, r121473);
        double r121477 = -r121476;
        double r121478 = z;
        double r121479 = r121477 - r121478;
        double r121480 = t;
        double r121481 = log(r121480);
        double r121482 = r121479 + r121481;
        return r121482;
}

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. Taylor expanded around inf 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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)))