Average Error: 9.0 → 0.4
Time: 31.7s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r17694575 = x;
        double r17694576 = y;
        double r17694577 = log(r17694576);
        double r17694578 = r17694575 * r17694577;
        double r17694579 = z;
        double r17694580 = 1.0;
        double r17694581 = r17694580 - r17694576;
        double r17694582 = log(r17694581);
        double r17694583 = r17694579 * r17694582;
        double r17694584 = r17694578 + r17694583;
        double r17694585 = t;
        double r17694586 = r17694584 - r17694585;
        return r17694586;
}
↓
double f(double x, double y, double z, double t) {
        double r17694587 = z;
        double r17694588 = 1.0;
        double r17694589 = log(r17694588);
        double r17694590 = y;
        double r17694591 = r17694590 * r17694588;
        double r17694592 = r17694589 - r17694591;
        double r17694593 = 0.5;
        double r17694594 = r17694588 / r17694590;
        double r17694595 = r17694593 / r17694594;
        double r17694596 = r17694595 / r17694594;
        double r17694597 = r17694592 - r17694596;
        double r17694598 = r17694587 * r17694597;
        double r17694599 = cbrt(r17694590);
        double r17694600 = r17694599 * r17694599;
        double r17694601 = log(r17694600);
        double r17694602 = x;
        double r17694603 = r17694601 * r17694602;
        double r17694604 = 0.3333333333333333;
        double r17694605 = pow(r17694590, r17694604);
        double r17694606 = log(r17694605);
        double r17694607 = r17694602 * r17694606;
        double r17694608 = r17694603 + r17694607;
        double r17694609 = r17694598 + r17694608;
        double r17694610 = t;
        double r17694611 = r17694609 - r17694610;
        return r17694611;
}

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

Original9.0
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.0

    \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.3

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.3

    \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]
  8. Using strategy rm

  9. Applied pow1/30.4

    \[\leadsto \left(\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \color{blue}{\left({y}^{\frac{1}{3}}\right)}\right) + z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

  10. Final simplification0.4

    \[\leadsto \left(z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019163 
(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) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))