Average Error: 9.7 → 0.4
Time: 26.7s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\sqrt[3]{{y}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left(\sqrt[3]{{y}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r322455 = x;
        double r322456 = y;
        double r322457 = log(r322456);
        double r322458 = r322455 * r322457;
        double r322459 = z;
        double r322460 = 1.0;
        double r322461 = r322460 - r322456;
        double r322462 = log(r322461);
        double r322463 = r322459 * r322462;
        double r322464 = r322458 + r322463;
        double r322465 = t;
        double r322466 = r322464 - r322465;
        return r322466;
}
↓
double f(double x, double y, double z, double t) {
        double r322467 = 2.0;
        double r322468 = y;
        double r322469 = 0.6666666666666666;
        double r322470 = pow(r322468, r322469);
        double r322471 = cbrt(r322470);
        double r322472 = cbrt(r322468);
        double r322473 = cbrt(r322472);
        double r322474 = r322471 * r322473;
        double r322475 = log(r322474);
        double r322476 = r322467 * r322475;
        double r322477 = x;
        double r322478 = r322476 * r322477;
        double r322479 = log(r322472);
        double r322480 = r322479 * r322477;
        double r322481 = r322478 + r322480;
        double r322482 = z;
        double r322483 = 1.0;
        double r322484 = log(r322483);
        double r322485 = r322483 * r322468;
        double r322486 = r322484 - r322485;
        double r322487 = r322482 * r322486;
        double r322488 = 0.5;
        double r322489 = pow(r322468, r322467);
        double r322490 = r322482 * r322489;
        double r322491 = pow(r322483, r322467);
        double r322492 = r322490 / r322491;
        double r322493 = r322488 * r322492;
        double r322494 = r322487 - r322493;
        double r322495 = r322481 + r322494;
        double r322496 = t;
        double r322497 = r322495 - r322496;
        return r322497;
}

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

Original9.7
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 9.7

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied add-cube-cbrt0.4

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

  12. Applied cbrt-prod0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +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))