Average Error: 9.1 → 0.4
Time: 8.5s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot x + \left(z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot x + \left(z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\right)
double f(double x, double y, double z, double t) {
        double r510397 = x;
        double r510398 = y;
        double r510399 = log(r510398);
        double r510400 = r510397 * r510399;
        double r510401 = z;
        double r510402 = 1.0;
        double r510403 = r510402 - r510398;
        double r510404 = log(r510403);
        double r510405 = r510401 * r510404;
        double r510406 = r510400 + r510405;
        double r510407 = t;
        double r510408 = r510406 - r510407;
        return r510408;
}
↓
double f(double x, double y, double z, double t) {
        double r510409 = x;
        double r510410 = y;
        double r510411 = cbrt(r510410);
        double r510412 = r510411 * r510411;
        double r510413 = log(r510412);
        double r510414 = r510409 * r510413;
        double r510415 = 0.3333333333333333;
        double r510416 = pow(r510410, r510415);
        double r510417 = log(r510416);
        double r510418 = r510417 * r510409;
        double r510419 = z;
        double r510420 = 1.0;
        double r510421 = log(r510420);
        double r510422 = r510420 * r510410;
        double r510423 = 0.5;
        double r510424 = 2.0;
        double r510425 = pow(r510410, r510424);
        double r510426 = pow(r510420, r510424);
        double r510427 = r510425 / r510426;
        double r510428 = r510423 * r510427;
        double r510429 = r510422 + r510428;
        double r510430 = r510421 - r510429;
        double r510431 = r510419 * r510430;
        double r510432 = t;
        double r510433 = r510431 - r510432;
        double r510434 = r510418 + r510433;
        double r510435 = r510414 + r510434;
        return r510435;
}

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

Original9.1
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.1

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

  2. Taylor expanded around 0 0.4

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

  4. Applied associate--l+0.4

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

  6. Applied add-cube-cbrt0.4

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

  7. Applied log-prod0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Applied associate-+l+0.4

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

  10. Simplified0.4

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

  12. Applied pow1/30.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 
(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))