Average Error: 1.9 → 0.4
Time: 33.2s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(\frac{z \cdot z}{1}, \frac{\frac{1}{2}}{1}, \mathsf{fma}\left(1, z, b\right)\right), a, y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(\frac{z \cdot z}{1}, \frac{\frac{1}{2}}{1}, \mathsf{fma}\left(1, z, b\right)\right), a, y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}
double f(double x, double y, double z, double t, double a, double b) {
        double r103293 = x;
        double r103294 = y;
        double r103295 = z;
        double r103296 = log(r103295);
        double r103297 = t;
        double r103298 = r103296 - r103297;
        double r103299 = r103294 * r103298;
        double r103300 = a;
        double r103301 = 1.0;
        double r103302 = r103301 - r103295;
        double r103303 = log(r103302);
        double r103304 = b;
        double r103305 = r103303 - r103304;
        double r103306 = r103300 * r103305;
        double r103307 = r103299 + r103306;
        double r103308 = exp(r103307);
        double r103309 = r103293 * r103308;
        return r103309;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r103310 = x;
        double r103311 = 1.0;
        double r103312 = log(r103311);
        double r103313 = z;
        double r103314 = r103313 * r103313;
        double r103315 = r103314 / r103311;
        double r103316 = 0.5;
        double r103317 = r103316 / r103311;
        double r103318 = b;
        double r103319 = fma(r103311, r103313, r103318);
        double r103320 = fma(r103315, r103317, r103319);
        double r103321 = r103312 - r103320;
        double r103322 = a;
        double r103323 = y;
        double r103324 = cbrt(r103313);
        double r103325 = log(r103324);
        double r103326 = t;
        double r103327 = r103325 - r103326;
        double r103328 = r103323 * r103327;
        double r103329 = fma(r103321, r103322, r103328);
        double r103330 = r103324 * r103324;
        double r103331 = log(r103330);
        double r103332 = r103331 * r103323;
        double r103333 = r103329 + r103332;
        double r103334 = exp(r103333);
        double r103335 = r103310 * r103334;
        return r103335;
}

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

Derivation

  1. Initial program 1.9

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(1 \cdot z + \frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}}\right)\right)} - b\right)}\]

  3. Simplified0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right)} - b\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - t\right) + a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right)}\]

  6. Applied log-prod0.5

    \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right) + a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right)}\]

  7. Applied associate--l+0.5

    \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)} + a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right)}\]

  8. Applied distribute-lft-in0.6

    \[\leadsto x \cdot e^{\color{blue}{\left(y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)} + a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right)}\]

  9. Applied associate-+l+0.6

    \[\leadsto x \cdot e^{\color{blue}{y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \left(y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right) + a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right)\right)}}\]

  10. Simplified0.4

    \[\leadsto x \cdot e^{y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \color{blue}{\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(\frac{z \cdot z}{1}, \frac{\frac{1}{2}}{1}, \mathsf{fma}\left(1, z, b\right)\right), a, \left(\log \left(\sqrt[3]{z}\right) - t\right) \cdot y\right)}}\]

  11. Final simplification0.4

    \[\leadsto x \cdot e^{\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(\frac{z \cdot z}{1}, \frac{\frac{1}{2}}{1}, \mathsf{fma}\left(1, z, b\right)\right), a, y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))