Average Error: 9.0 → 0.4
Time: 22.5s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r307146 = x;
        double r307147 = y;
        double r307148 = log(r307147);
        double r307149 = r307146 * r307148;
        double r307150 = z;
        double r307151 = 1.0;
        double r307152 = r307151 - r307147;
        double r307153 = log(r307152);
        double r307154 = r307150 * r307153;
        double r307155 = r307149 + r307154;
        double r307156 = t;
        double r307157 = r307155 - r307156;
        return r307157;
}
↓
double f(double x, double y, double z, double t) {
        double r307158 = 2.0;
        double r307159 = y;
        double r307160 = sqrt(r307159);
        double r307161 = cbrt(r307160);
        double r307162 = r307161 * r307161;
        double r307163 = log(r307162);
        double r307164 = r307158 * r307163;
        double r307165 = x;
        double r307166 = r307164 * r307165;
        double r307167 = 1.0;
        double r307168 = r307167 / r307159;
        double r307169 = -0.3333333333333333;
        double r307170 = pow(r307168, r307169);
        double r307171 = log(r307170);
        double r307172 = r307171 * r307165;
        double r307173 = r307166 + r307172;
        double r307174 = z;
        double r307175 = 1.0;
        double r307176 = log(r307175);
        double r307177 = 0.5;
        double r307178 = r307175 * r307175;
        double r307179 = r307178 / r307159;
        double r307180 = r307177 / r307179;
        double r307181 = r307175 + r307180;
        double r307182 = r307159 * r307181;
        double r307183 = r307176 - r307182;
        double r307184 = r307174 * r307183;
        double r307185 = r307173 + r307184;
        double r307186 = t;
        double r307187 = r307185 - r307186;
        return r307187;
}

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

Derivation

  1. Initial program 9.0

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Taylor expanded around inf 0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied cbrt-prod0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 
(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))