Average Error: 9.6 → 0.4
Time: 20.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + \left(\left(\frac{y}{1} \cdot y\right) \cdot \frac{\frac{-1}{2}}{1} + \left(\log 1 - y \cdot 1\right)\right) \cdot z\right) + \log \left({y}^{\frac{2}{3}}\right) \cdot x\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + \left(\left(\frac{y}{1} \cdot y\right) \cdot \frac{\frac{-1}{2}}{1} + \left(\log 1 - y \cdot 1\right)\right) \cdot z\right) + \log \left({y}^{\frac{2}{3}}\right) \cdot x\right) - t
double f(double x, double y, double z, double t) {
        double r379271 = x;
        double r379272 = y;
        double r379273 = log(r379272);
        double r379274 = r379271 * r379273;
        double r379275 = z;
        double r379276 = 1.0;
        double r379277 = r379276 - r379272;
        double r379278 = log(r379277);
        double r379279 = r379275 * r379278;
        double r379280 = r379274 + r379279;
        double r379281 = t;
        double r379282 = r379280 - r379281;
        return r379282;
}
↓
double f(double x, double y, double z, double t) {
        double r379283 = x;
        double r379284 = y;
        double r379285 = cbrt(r379284);
        double r379286 = log(r379285);
        double r379287 = r379283 * r379286;
        double r379288 = 1.0;
        double r379289 = r379284 / r379288;
        double r379290 = r379289 * r379284;
        double r379291 = -0.5;
        double r379292 = r379291 / r379288;
        double r379293 = r379290 * r379292;
        double r379294 = log(r379288);
        double r379295 = r379284 * r379288;
        double r379296 = r379294 - r379295;
        double r379297 = r379293 + r379296;
        double r379298 = z;
        double r379299 = r379297 * r379298;
        double r379300 = r379287 + r379299;
        double r379301 = 0.6666666666666666;
        double r379302 = pow(r379284, r379301);
        double r379303 = log(r379302);
        double r379304 = r379303 * r379283;
        double r379305 = r379300 + r379304;
        double r379306 = t;
        double r379307 = r379305 - r379306;
        return r379307;
}

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

Original9.6
Target0.2
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 9.6

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-rgt-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Simplified0.4

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

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

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

  12. Applied cbrt-prod0.4

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

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

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

  14. Applied cbrt-prod0.4

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

  15. Applied swap-sqr0.4

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

  16. Simplified0.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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))