Average Error: 9.6 → 0.3
Time: 8.3s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\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 fma(log(y), x, (fma(z, log(1.0), -fma(1.0, (z * y), (0.5 * ((z * 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

Original9.6
Target0.3
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, z \cdot \log \left(1 - y\right) - t\right)}\]

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(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))