Average Error: 9.0 → 0.4
Time: 22.7s
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 r341026 = x;
        double r341027 = y;
        double r341028 = log(r341027);
        double r341029 = r341026 * r341028;
        double r341030 = z;
        double r341031 = 1.0;
        double r341032 = r341031 - r341027;
        double r341033 = log(r341032);
        double r341034 = r341030 * r341033;
        double r341035 = r341029 + r341034;
        double r341036 = t;
        double r341037 = r341035 - r341036;
        return r341037;
}
↓
double f(double x, double y, double z, double t) {
        double r341038 = x;
        double r341039 = y;
        double r341040 = log(r341039);
        double r341041 = z;
        double r341042 = 1.0;
        double r341043 = log(r341042);
        double r341044 = 0.5;
        double r341045 = r341042 * r341042;
        double r341046 = r341045 / r341039;
        double r341047 = r341044 / r341046;
        double r341048 = r341042 + r341047;
        double r341049 = r341039 * r341048;
        double r341050 = r341043 - r341049;
        double r341051 = r341041 * r341050;
        double r341052 = fma(r341038, r341040, r341051);
        double r341053 = t;
        double r341054 = r341052 - r341053;
        return r341054;
}

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))