Average Error: 9.9 → 0.3
Time: 24.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({y}^{\frac{1}{3}}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({y}^{\frac{1}{3}}\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 r367486 = x;
        double r367487 = y;
        double r367488 = log(r367487);
        double r367489 = r367486 * r367488;
        double r367490 = z;
        double r367491 = 1.0;
        double r367492 = r367491 - r367487;
        double r367493 = log(r367492);
        double r367494 = r367490 * r367493;
        double r367495 = r367489 + r367494;
        double r367496 = t;
        double r367497 = r367495 - r367496;
        return r367497;
}
↓
double f(double x, double y, double z, double t) {
        double r367498 = x;
        double r367499 = 2.0;
        double r367500 = y;
        double r367501 = cbrt(r367500);
        double r367502 = log(r367501);
        double r367503 = r367499 * r367502;
        double r367504 = r367498 * r367503;
        double r367505 = 0.3333333333333333;
        double r367506 = pow(r367500, r367505);
        double r367507 = log(r367506);
        double r367508 = r367507 * r367498;
        double r367509 = r367504 + r367508;
        double r367510 = z;
        double r367511 = 1.0;
        double r367512 = log(r367511);
        double r367513 = r367511 * r367500;
        double r367514 = r367512 - r367513;
        double r367515 = r367510 * r367514;
        double r367516 = 0.5;
        double r367517 = pow(r367500, r367499);
        double r367518 = r367510 * r367517;
        double r367519 = pow(r367511, r367499);
        double r367520 = r367518 / r367519;
        double r367521 = r367516 * r367520;
        double r367522 = r367515 - r367521;
        double r367523 = r367509 + r367522;
        double r367524 = t;
        double r367525 = r367523 - r367524;
        return r367525;
}

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

Original9.9
Target0.2
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.9

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

  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

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

  9. Simplified0.4

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

  11. Applied pow1/30.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(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) (* (/ 0.333333333333333315 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))