Average Error: 0.1 → 0.1
Time: 21.3s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\log t + \left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left(\sqrt[3]{y}\right) \cdot x\right) - y\right) - z\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\log t + \left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left(\sqrt[3]{y}\right) \cdot x\right) - y\right) - z\right)
double f(double x, double y, double z, double t) {
        double r82441 = x;
        double r82442 = y;
        double r82443 = log(r82442);
        double r82444 = r82441 * r82443;
        double r82445 = r82444 - r82442;
        double r82446 = z;
        double r82447 = r82445 - r82446;
        double r82448 = t;
        double r82449 = log(r82448);
        double r82450 = r82447 + r82449;
        return r82450;
}
↓
double f(double x, double y, double z, double t) {
        double r82451 = t;
        double r82452 = log(r82451);
        double r82453 = x;
        double r82454 = 2.0;
        double r82455 = y;
        double r82456 = cbrt(r82455);
        double r82457 = log(r82456);
        double r82458 = r82454 * r82457;
        double r82459 = r82457 * r82453;
        double r82460 = fma(r82453, r82458, r82459);
        double r82461 = r82460 - r82455;
        double r82462 = z;
        double r82463 = r82461 - r82462;
        double r82464 = r82452 + r82463;
        return r82464;
}

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-cube-cbrt0.1

    \[\leadsto \left(\left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} - y\right) - z\right) + \log t\]

  4. Applied log-prod0.1

    \[\leadsto \left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

  5. Applied distribute-lft-in0.1

    \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

  6. Simplified0.1

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]
  7. Using strategy rm

  8. Applied fma-def0.1

    \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

  9. Final simplification0.1

    \[\leadsto \log t + \left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left(\sqrt[3]{y}\right) \cdot x\right) - y\right) - z\right)\]

Reproduce

herbie shell --seed 2019179 +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)))