Average Error: 9.6 → 0.6
Time: 28.1s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - y \cdot 1.0\right), z, \left(x \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y \cdot \log y} - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t
↓
\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - y \cdot 1.0\right), z, \left(x \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y \cdot \log y} - t\right)
double f(double x, double y, double z, double t) {
        double r16138341 = x;
        double r16138342 = y;
        double r16138343 = log(r16138342);
        double r16138344 = r16138341 * r16138343;
        double r16138345 = z;
        double r16138346 = 1.0;
        double r16138347 = r16138346 - r16138342;
        double r16138348 = log(r16138347);
        double r16138349 = r16138345 * r16138348;
        double r16138350 = r16138344 + r16138349;
        double r16138351 = t;
        double r16138352 = r16138350 - r16138351;
        return r16138352;
}
↓
double f(double x, double y, double z, double t) {
        double r16138353 = y;
        double r16138354 = 1.0;
        double r16138355 = r16138353 / r16138354;
        double r16138356 = r16138355 * r16138355;
        double r16138357 = -0.5;
        double r16138358 = log(r16138354);
        double r16138359 = r16138353 * r16138354;
        double r16138360 = r16138358 - r16138359;
        double r16138361 = fma(r16138356, r16138357, r16138360);
        double r16138362 = z;
        double r16138363 = x;
        double r16138364 = log(r16138353);
        double r16138365 = cbrt(r16138364);
        double r16138366 = r16138363 * r16138365;
        double r16138367 = r16138364 * r16138364;
        double r16138368 = cbrt(r16138367);
        double r16138369 = r16138366 * r16138368;
        double r16138370 = t;
        double r16138371 = r16138369 - r16138370;
        double r16138372 = fma(r16138361, r16138362, r16138371);
        return r16138372;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.3
Herbie0.6
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

    \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Simplified9.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1.0 - y\right), z, \log y \cdot x - t\right)}\]

  3. Taylor expanded around 0 0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)}, z, \log y \cdot x - t\right)\]

  4. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - 1.0 \cdot y\right)}, z, \log y \cdot x - t\right)\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.8

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - 1.0 \cdot y\right), z, \color{blue}{\left(\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y}\right)} \cdot x - t\right)\]

  7. Applied associate-*l*0.8

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - 1.0 \cdot y\right), z, \color{blue}{\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \left(\sqrt[3]{\log y} \cdot x\right)} - t\right)\]
  8. Using strategy rm

  9. Applied cbrt-unprod0.6

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - 1.0 \cdot y\right), z, \color{blue}{\sqrt[3]{\log y \cdot \log y}} \cdot \left(\sqrt[3]{\log y} \cdot x\right) - t\right)\]

  10. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - y \cdot 1.0\right), z, \left(x \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y \cdot \log y} - t\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))