Average Error: 0.1 → 0.1
Time: 20.8s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\mathsf{fma}\left(\log y, x, \left(-y\right) - z\right) + \log \left(\sqrt{t}\right)\right) + \log \left(\sqrt{t}\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\mathsf{fma}\left(\log y, x, \left(-y\right) - z\right) + \log \left(\sqrt{t}\right)\right) + \log \left(\sqrt{t}\right)
double f(double x, double y, double z, double t) {
        double r93016 = x;
        double r93017 = y;
        double r93018 = log(r93017);
        double r93019 = r93016 * r93018;
        double r93020 = r93019 - r93017;
        double r93021 = z;
        double r93022 = r93020 - r93021;
        double r93023 = t;
        double r93024 = log(r93023);
        double r93025 = r93022 + r93024;
        return r93025;
}
↓
double f(double x, double y, double z, double t) {
        double r93026 = y;
        double r93027 = log(r93026);
        double r93028 = x;
        double r93029 = -r93026;
        double r93030 = z;
        double r93031 = r93029 - r93030;
        double r93032 = fma(r93027, r93028, r93031);
        double r93033 = t;
        double r93034 = sqrt(r93033);
        double r93035 = log(r93034);
        double r93036 = r93032 + r93035;
        double r93037 = r93036 + r93035;
        return r93037;
}

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 add-sqr-sqrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied associate-+r+0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  (+ (- (- (* x (log y)) y) z) (log t)))