Average Error: 9.4 → 1.2
Time: 8.1s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.86976225029446352 \cdot 10^{94}:\\ \;\;\;\;\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \log \left(1 - y\right)\right) - t\\ \mathbf{elif}\;x \le 5.588580349961717 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt[3]{\left(x \cdot x\right) \cdot {\left(\log y\right)}^{2}} \cdot \sqrt[3]{x \cdot \log y} + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \log y\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\\ \end{array}\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.86976225029446352 \cdot 10^{94}:\\
\;\;\;\;\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \log \left(1 - y\right)\right) - t\\

\mathbf{elif}\;x \le 5.588580349961717 \cdot 10^{-310}:\\
\;\;\;\;\left(\sqrt[3]{\left(x \cdot x\right) \cdot {\left(\log y\right)}^{2}} \cdot \sqrt[3]{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\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \log y\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\\

\end{array}
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) {
	double VAR;
	if ((x <= -5.8697622502944635e+94)) {
		VAR = ((double) (((double) (((double) (((double) (x * ((double) (2.0 * ((double) log(((double) cbrt(y)))))))) + ((double) (x * ((double) log(((double) cbrt(y)))))))) + ((double) (z * ((double) log(((double) (1.0 - y)))))))) - t));
	} else {
		double VAR_1;
		if ((x <= 5.5885803499617e-310)) {
			VAR_1 = ((double) (((double) (((double) (((double) cbrt(((double) (((double) (x * x)) * ((double) pow(((double) log(y)), 2.0)))))) * ((double) cbrt(((double) (x * ((double) log(y)))))))) + ((double) (z * ((double) (((double) log(1.0)) - ((double) (((double) (1.0 * y)) + ((double) (0.5 * ((double) (((double) pow(y, 2.0)) / ((double) pow(1.0, 2.0)))))))))))))) - t));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) sqrt(x)) * ((double) (((double) sqrt(x)) * ((double) log(y)))))) + ((double) (z * ((double) (((double) log(1.0)) - ((double) (((double) (1.0 * y)) + ((double) (0.5 * ((double) (((double) pow(y, 2.0)) / ((double) pow(1.0, 2.0)))))))))))))) - t));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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
Herbie1.2
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.8697622502944635e+94

    1. Initial program 2.1

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
    2. Using strategy rm

    3. Applied add-cube-cbrt2.1

      \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \log \left(1 - y\right)\right) - t\]

    4. Applied log-prod2.2

      \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \log \left(1 - y\right)\right) - t\]

    5. Applied distribute-lft-in2.2

      \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \log \left(1 - y\right)\right) - t\]

    6. Simplified2.2

      \[\leadsto \left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \log \left(1 - y\right)\right) - t\]

    if -5.8697622502944635e+94 < x < 5.5885803499617e-310

    1. Initial program 13.8

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

    2. Taylor expanded around 0 0.5

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

    4. Applied add-cube-cbrt0.7

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

    6. Applied cbrt-unprod1.8

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

    7. Simplified1.9

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

    if 5.5885803499617e-310 < x

    1. Initial program 9.3

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

    2. Taylor expanded around 0 0.3

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

    4. Applied add-sqr-sqrt0.5

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

    5. Applied associate-*l*0.4

      \[\leadsto \left(\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot \log y\right)} + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.86976225029446352 \cdot 10^{94}:\\ \;\;\;\;\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \log \left(1 - y\right)\right) - t\\ \mathbf{elif}\;x \le 5.588580349961717 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt[3]{\left(x \cdot x\right) \cdot {\left(\log y\right)}^{2}} \cdot \sqrt[3]{x \cdot \log y} + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \log y\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(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))