Average Error: 9.1 → 0.4
Time: 23.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1 \cdot y, z, \frac{\frac{z}{1} \cdot \frac{1}{2}}{\frac{\frac{1}{y}}{y}}\right)\right) - t\right)\]
\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 \cdot y, z, \frac{\frac{z}{1} \cdot \frac{1}{2}}{\frac{\frac{1}{y}}{y}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r324558 = x;
        double r324559 = y;
        double r324560 = log(r324559);
        double r324561 = r324558 * r324560;
        double r324562 = z;
        double r324563 = 1.0;
        double r324564 = r324563 - r324559;
        double r324565 = log(r324564);
        double r324566 = r324562 * r324565;
        double r324567 = r324561 + r324566;
        double r324568 = t;
        double r324569 = r324567 - r324568;
        return r324569;
}
↓
double f(double x, double y, double z, double t) {
        double r324570 = y;
        double r324571 = log(r324570);
        double r324572 = x;
        double r324573 = z;
        double r324574 = 1.0;
        double r324575 = log(r324574);
        double r324576 = r324574 * r324570;
        double r324577 = r324573 / r324574;
        double r324578 = 0.5;
        double r324579 = r324577 * r324578;
        double r324580 = r324574 / r324570;
        double r324581 = r324580 / r324570;
        double r324582 = r324579 / r324581;
        double r324583 = fma(r324576, r324573, r324582);
        double r324584 = -r324583;
        double r324585 = fma(r324573, r324575, r324584);
        double r324586 = t;
        double r324587 = r324585 - r324586;
        double r324588 = fma(r324571, r324572, r324587);
        return r324588;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.1
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 9.1

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied *-un-lft-identity0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied distribute-lft-out--0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))