Average Error: 9.1 → 0.4
Time: 8.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[x \cdot \log y + \left(\left(z \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
x \cdot \log y + \left(\left(z \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right) - t\right)
double code(double x, double y, double z, double t) {
	return (((x * log(y)) + (z * log((1.0 - y)))) - t);
}
↓
double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (((z * (sqrt(log(1.0)) + sqrt(((1.0 * y) + (0.5 * (pow(y, 2.0) / pow(1.0, 2.0))))))) * (sqrt(log(1.0)) - sqrt(((1.0 * y) + (0.5 * (pow(y, 2.0) / pow(1.0, 2.0))))))) - 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

Original9.1
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 9.1

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

  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. Using strategy rm

  4. Applied associate--l+0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied difference-of-squares0.4

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

  9. Applied associate-*r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 
(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))