Average Error: 9.6 → 0.4
Time: 8.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)
double f(double x, double y, double z, double t) {
        double r422523 = x;
        double r422524 = y;
        double r422525 = log(r422524);
        double r422526 = r422523 * r422525;
        double r422527 = z;
        double r422528 = 1.0;
        double r422529 = r422528 - r422524;
        double r422530 = log(r422529);
        double r422531 = r422527 * r422530;
        double r422532 = r422526 + r422531;
        double r422533 = t;
        double r422534 = r422532 - r422533;
        return r422534;
}
↓
double f(double x, double y, double z, double t) {
        double r422535 = y;
        double r422536 = log(r422535);
        double r422537 = x;
        double r422538 = z;
        double r422539 = 1.0;
        double r422540 = log(r422539);
        double r422541 = r422539 * r422535;
        double r422542 = 0.5;
        double r422543 = 2.0;
        double r422544 = pow(r422535, r422543);
        double r422545 = pow(r422539, r422543);
        double r422546 = r422544 / r422545;
        double r422547 = r422542 * r422546;
        double r422548 = r422541 + r422547;
        double r422549 = r422540 - r422548;
        double r422550 = t;
        double r422551 = -r422550;
        double r422552 = fma(r422538, r422549, r422551);
        double r422553 = fma(r422536, r422537, r422552);
        return r422553;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.6

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

  3. Taylor expanded around 0 0.4

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

  5. Applied fma-neg0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 +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))