Average Error: 9.2 → 0.3
Time: 20.8s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x, \log y, z \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(x, \log y, z \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r318300 = x;
        double r318301 = y;
        double r318302 = log(r318301);
        double r318303 = r318300 * r318302;
        double r318304 = z;
        double r318305 = 1.0;
        double r318306 = r318305 - r318301;
        double r318307 = log(r318306);
        double r318308 = r318304 * r318307;
        double r318309 = r318303 + r318308;
        double r318310 = t;
        double r318311 = r318309 - r318310;
        return r318311;
}
↓
double f(double x, double y, double z, double t) {
        double r318312 = x;
        double r318313 = y;
        double r318314 = log(r318313);
        double r318315 = z;
        double r318316 = 1.0;
        double r318317 = log(r318316);
        double r318318 = 0.5;
        double r318319 = 2.0;
        double r318320 = pow(r318313, r318319);
        double r318321 = pow(r318316, r318319);
        double r318322 = r318320 / r318321;
        double r318323 = r318318 * r318322;
        double r318324 = fma(r318316, r318313, r318323);
        double r318325 = r318317 - r318324;
        double r318326 = r318315 * r318325;
        double r318327 = fma(r318312, r318314, r318326);
        double r318328 = t;
        double r318329 = r318327 - r318328;
        return r318329;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.2
Target0.3
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.2

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

  2. Simplified9.2

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 +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) (* (/ 0.333333333333333315 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

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