Average Error: 9.4 → 1.4
Time: 8.2s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), z, \left(\sqrt[3]{x \cdot \log y - t} \cdot \sqrt[3]{x \cdot \log y - t}\right) \cdot \sqrt[3]{x \cdot \log y - t}\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), z, \left(\sqrt[3]{x \cdot \log y - t} \cdot \sqrt[3]{x \cdot \log y - t}\right) \cdot \sqrt[3]{x \cdot \log y - t}\right)
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(y)))) + ((double) (z * ((double) log(((double) (1.0 - y)))))))) - t));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) fma(((double) (((double) log(1.0)) - ((double) fma(1.0, y, ((double) (0.5 * ((double) (((double) pow(y, 2.0)) / ((double) pow(1.0, 2.0)))))))))), z, ((double) (((double) (((double) cbrt(((double) (((double) (x * ((double) log(y)))) - t)))) * ((double) cbrt(((double) (((double) (x * ((double) log(y)))) - t)))))) * ((double) cbrt(((double) (((double) (x * ((double) log(y)))) - t))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.3
Herbie1.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.4

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

  2. Taylor expanded around 0 0.4

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

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

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

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

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

  6. Applied distribute-lft-out--0.4

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

  7. Simplified0.4

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

  9. Applied add-cube-cbrt1.4

    \[\leadsto 1 \cdot \mathsf{fma}\left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), z, \color{blue}{\left(\sqrt[3]{x \cdot \log y - t} \cdot \sqrt[3]{x \cdot \log y - t}\right) \cdot \sqrt[3]{x \cdot \log y - t}}\right)\]

  10. Final simplification1.4

    \[\leadsto \mathsf{fma}\left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), z, \left(\sqrt[3]{x \cdot \log y - t} \cdot \sqrt[3]{x \cdot \log y - t}\right) \cdot \sqrt[3]{x \cdot \log y - t}\right)\]

Reproduce

herbie shell --seed 2020121 +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))