Average Error: 9.6 → 0.4
Time: 28.7s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \mathsf{fma}\left(x, \log \left(\sqrt[3]{y}\right), z \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \mathsf{fma}\left(x, \log \left(\sqrt[3]{y}\right), z \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r285311 = x;
        double r285312 = y;
        double r285313 = log(r285312);
        double r285314 = r285311 * r285313;
        double r285315 = z;
        double r285316 = 1.0;
        double r285317 = r285316 - r285312;
        double r285318 = log(r285317);
        double r285319 = r285315 * r285318;
        double r285320 = r285314 + r285319;
        double r285321 = t;
        double r285322 = r285320 - r285321;
        return r285322;
}
↓
double f(double x, double y, double z, double t) {
        double r285323 = x;
        double r285324 = y;
        double r285325 = cbrt(r285324);
        double r285326 = r285325 * r285325;
        double r285327 = log(r285326);
        double r285328 = r285323 * r285327;
        double r285329 = log(r285325);
        double r285330 = z;
        double r285331 = 1.0;
        double r285332 = log(r285331);
        double r285333 = 0.5;
        double r285334 = 2.0;
        double r285335 = pow(r285324, r285334);
        double r285336 = pow(r285331, r285334);
        double r285337 = r285335 / r285336;
        double r285338 = r285333 * r285337;
        double r285339 = fma(r285331, r285324, r285338);
        double r285340 = r285332 - r285339;
        double r285341 = r285330 * r285340;
        double r285342 = fma(r285323, r285329, r285341);
        double r285343 = r285328 + r285342;
        double r285344 = t;
        double r285345 = r285343 - r285344;
        return r285345;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))