Average Error: 9.7 → 0.3
Time: 30.4s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\log \left(\sqrt{y}\right) \cdot x + \log \left(\sqrt{y}\right) \cdot x\right) + \left(\left(\log 1 - y \cdot 1\right) \cdot z - \frac{z}{\frac{1}{y}} \cdot \frac{\frac{1}{2}}{\frac{1}{y}}\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\log \left(\sqrt{y}\right) \cdot x + \log \left(\sqrt{y}\right) \cdot x\right) + \left(\left(\log 1 - y \cdot 1\right) \cdot z - \frac{z}{\frac{1}{y}} \cdot \frac{\frac{1}{2}}{\frac{1}{y}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r18407794 = x;
        double r18407795 = y;
        double r18407796 = log(r18407795);
        double r18407797 = r18407794 * r18407796;
        double r18407798 = z;
        double r18407799 = 1.0;
        double r18407800 = r18407799 - r18407795;
        double r18407801 = log(r18407800);
        double r18407802 = r18407798 * r18407801;
        double r18407803 = r18407797 + r18407802;
        double r18407804 = t;
        double r18407805 = r18407803 - r18407804;
        return r18407805;
}
↓
double f(double x, double y, double z, double t) {
        double r18407806 = y;
        double r18407807 = sqrt(r18407806);
        double r18407808 = log(r18407807);
        double r18407809 = x;
        double r18407810 = r18407808 * r18407809;
        double r18407811 = r18407810 + r18407810;
        double r18407812 = 1.0;
        double r18407813 = log(r18407812);
        double r18407814 = r18407806 * r18407812;
        double r18407815 = r18407813 - r18407814;
        double r18407816 = z;
        double r18407817 = r18407815 * r18407816;
        double r18407818 = r18407812 / r18407806;
        double r18407819 = r18407816 / r18407818;
        double r18407820 = 0.5;
        double r18407821 = r18407820 / r18407818;
        double r18407822 = r18407819 * r18407821;
        double r18407823 = r18407817 - r18407822;
        double r18407824 = r18407811 + r18407823;
        double r18407825 = t;
        double r18407826 = r18407824 - r18407825;
        return r18407826;
}

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

Derivation

  1. Initial program 9.7

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied log-prod0.3

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

  7. Applied distribute-lft-in0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(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) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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