Average Error: 9.6 → 0.3
Time: 8.0s
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 r524030 = x;
        double r524031 = y;
        double r524032 = log(r524031);
        double r524033 = r524030 * r524032;
        double r524034 = z;
        double r524035 = 1.0;
        double r524036 = r524035 - r524031;
        double r524037 = log(r524036);
        double r524038 = r524034 * r524037;
        double r524039 = r524033 + r524038;
        double r524040 = t;
        double r524041 = r524039 - r524040;
        return r524041;
}
↓
double f(double x, double y, double z, double t) {
        double r524042 = y;
        double r524043 = log(r524042);
        double r524044 = x;
        double r524045 = z;
        double r524046 = 1.0;
        double r524047 = log(r524046);
        double r524048 = r524046 * r524042;
        double r524049 = 0.5;
        double r524050 = 2.0;
        double r524051 = pow(r524042, r524050);
        double r524052 = pow(r524046, r524050);
        double r524053 = r524051 / r524052;
        double r524054 = r524049 * r524053;
        double r524055 = r524048 + r524054;
        double r524056 = r524047 - r524055;
        double r524057 = t;
        double r524058 = -r524057;
        double r524059 = fma(r524045, r524056, r524058);
        double r524060 = fma(r524043, r524044, r524059);
        return r524060;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.3
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 9.6

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

  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 0.3

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

  5. Applied fma-neg0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020083 +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))