Average Error: 9.0 → 0.3
Time: 7.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\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, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r437607 = x;
        double r437608 = y;
        double r437609 = log(r437608);
        double r437610 = r437607 * r437609;
        double r437611 = z;
        double r437612 = 1.0;
        double r437613 = r437612 - r437608;
        double r437614 = log(r437613);
        double r437615 = r437611 * r437614;
        double r437616 = r437610 + r437615;
        double r437617 = t;
        double r437618 = r437616 - r437617;
        return r437618;
}
↓
double f(double x, double y, double z, double t) {
        double r437619 = y;
        double r437620 = log(r437619);
        double r437621 = x;
        double r437622 = z;
        double r437623 = 1.0;
        double r437624 = log(r437623);
        double r437625 = r437622 * r437619;
        double r437626 = 0.5;
        double r437627 = 2.0;
        double r437628 = pow(r437619, r437627);
        double r437629 = r437622 * r437628;
        double r437630 = pow(r437623, r437627);
        double r437631 = r437629 / r437630;
        double r437632 = r437626 * r437631;
        double r437633 = fma(r437623, r437625, r437632);
        double r437634 = -r437633;
        double r437635 = fma(r437622, r437624, r437634);
        double r437636 = t;
        double r437637 = r437635 - r437636;
        double r437638 = fma(r437620, r437621, r437637);
        return r437638;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.0
Target0.2
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.0

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

  2. Simplified9.0

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +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))