Average Error: 9.4 → 0.4
Time: 5.3s
Precision: binary64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - y \cdot \left(1 + y \cdot \frac{\frac{1}{2}}{1 \cdot 1}\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - y \cdot \left(1 + y \cdot \frac{\frac{1}{2}}{1 \cdot 1}\right)\right)\right)\right) - t
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(y)))) + ((double) (z * ((double) log(((double) (1.0 - y)))))))) - t));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(((double) (((double) pow(((double) cbrt(y)), 1.6666666666666667)) * ((double) cbrt(((double) cbrt(y)))))))))) + ((double) (((double) (x * ((double) log(((double) pow(y, 0.3333333333333333)))))) + ((double) (z * ((double) (((double) log(1.0)) - ((double) (y * ((double) (1.0 + ((double) (y * ((double) (0.5 / ((double) (1.0 * 1.0)))))))))))))))))) - t));
}

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

Original9.4
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 9.4

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

  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

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Simplified0.4

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

  11. Applied pow1/30.4

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

  13. Applied add-cube-cbrt0.4

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

  14. Applied associate-*r*0.4

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

  15. Simplified0.4

    \[\leadsto \left(x \cdot \log \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{\frac{5}{3}}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - y \cdot \left(1 + y \cdot \frac{\frac{1}{2}}{1 \cdot 1}\right)\right)\right)\right) - t\]

  16. Final simplification0.4

    \[\leadsto \left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - y \cdot \left(1 + y \cdot \frac{\frac{1}{2}}{1 \cdot 1}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (neg 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))