Average Error: 6.7 → 0.1
Time: 9.6s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
↓
\[\mathsf{fma}\left(-1, x \cdot \log \left(\frac{1}{y}\right), \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), -1 + z, -\log y\right)\right) - t \]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(-1, x \cdot \log \left(\frac{1}{y}\right), \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), -1 + z, -\log y\right)\right) - t

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))
↓
(FPCore (x y z t)
 :precision binary64
 (-
  (fma -1.0 (* x (log (/ 1.0 y))) (fma (log1p (- y)) (+ -1.0 z) (- (log y))))
  t))
double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log(1.0 - y))) - t;
}
↓
double code(double x, double y, double z, double t) {
	return fma(-1.0, (x * log(1.0 / y)), fma(log1p(-y), (-1.0 + z), -log(y))) - t;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.7

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \mathsf{log1p}\left(-y\right)\right) - t} \]

  3. Taylor expanded in x around 0 6.7

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

  4. Simplified0.1

    \[\leadsto \color{blue}{\left(\log y \cdot x + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z + -1, -\log y\right)\right)} - t \]

  5. Taylor expanded in y around inf 0.1

    \[\leadsto \left(\color{blue}{-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)} + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z + -1, -\log y\right)\right) - t \]

  6. Applied fma-def_binary640.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x \cdot \log \left(\frac{1}{y}\right), \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z + -1, -\log y\right)\right)} - t \]

  7. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(-1, x \cdot \log \left(\frac{1}{y}\right), \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), -1 + z, -\log y\right)\right) - t \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))