Average Error: 10.0 → 0.4
Time: 9.3s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\log \left(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) \cdot x + \left(\log \left(\sqrt[3]{y}\right) \cdot x + \left(z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\log \left(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) \cdot x + \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)
double f(double x, double y, double z, double t) {
        double r412455 = x;
        double r412456 = y;
        double r412457 = log(r412456);
        double r412458 = r412455 * r412457;
        double r412459 = z;
        double r412460 = 1.0;
        double r412461 = r412460 - r412456;
        double r412462 = log(r412461);
        double r412463 = r412459 * r412462;
        double r412464 = r412458 + r412463;
        double r412465 = t;
        double r412466 = r412464 - r412465;
        return r412466;
}
↓
double f(double x, double y, double z, double t) {
        double r412467 = y;
        double r412468 = cbrt(r412467);
        double r412469 = 0.3333333333333333;
        double r412470 = pow(r412467, r412469);
        double r412471 = r412468 * r412470;
        double r412472 = log(r412471);
        double r412473 = x;
        double r412474 = r412472 * r412473;
        double r412475 = log(r412468);
        double r412476 = r412475 * r412473;
        double r412477 = z;
        double r412478 = 1.0;
        double r412479 = log(r412478);
        double r412480 = r412478 * r412467;
        double r412481 = 0.5;
        double r412482 = 2.0;
        double r412483 = pow(r412467, r412482);
        double r412484 = pow(r412478, r412482);
        double r412485 = r412483 / r412484;
        double r412486 = r412481 * r412485;
        double r412487 = r412480 + r412486;
        double r412488 = r412479 - r412487;
        double r412489 = r412477 * r412488;
        double r412490 = t;
        double r412491 = r412489 - r412490;
        double r412492 = r412476 + r412491;
        double r412493 = r412474 + r412492;
        return r412493;
}

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

Original10.0
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 10.0

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

  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 associate--l+0.3

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

  6. Applied add-cube-cbrt0.3

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

  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-rgt-in0.4

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

  9. Applied associate-+l+0.4

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

  11. Applied pow1/30.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 
(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))