Average Error: 9.0 → 0.4
Time: 24.0s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x, \log y, z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]
\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 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r295172 = x;
        double r295173 = y;
        double r295174 = log(r295173);
        double r295175 = r295172 * r295174;
        double r295176 = z;
        double r295177 = 1.0;
        double r295178 = r295177 - r295173;
        double r295179 = log(r295178);
        double r295180 = r295176 * r295179;
        double r295181 = r295175 + r295180;
        double r295182 = t;
        double r295183 = r295181 - r295182;
        return r295183;
}
↓
double f(double x, double y, double z, double t) {
        double r295184 = x;
        double r295185 = y;
        double r295186 = log(r295185);
        double r295187 = z;
        double r295188 = 1.0;
        double r295189 = log(r295188);
        double r295190 = 0.5;
        double r295191 = r295188 * r295188;
        double r295192 = r295191 / r295185;
        double r295193 = r295190 / r295192;
        double r295194 = r295188 + r295193;
        double r295195 = r295185 * r295194;
        double r295196 = r295189 - r295195;
        double r295197 = r295187 * r295196;
        double r295198 = fma(r295184, r295186, r295197);
        double r295199 = t;
        double r295200 = r295198 - r295199;
        return r295200;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.0
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 9.0

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

  2. Simplified9.0

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +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))