Average Error: 7.2 → 0.1
Time: 6.5s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
↓
\[-\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), 1 - z, \mathsf{fma}\left(-\log y, x, \log y + t\right)\right) \]
(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 (log1p (- y)) (- 1.0 z) (fma (- (log y)) x (+ (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(log1p(-y), (1.0 - z), fma(-log(y), x, (log(y) + t)));
}

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end
↓
function code(x, y, z, t)
	return Float64(-fma(log1p(Float64(-y)), Float64(1.0 - z), fma(Float64(-log(y)), x, Float64(log(y) + t))))
end

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]
↓
code[x_, y_, z_, t_] := (-N[(N[Log[1 + (-y)], $MachinePrecision] * N[(1.0 - z), $MachinePrecision] + N[((-N[Log[y], $MachinePrecision]) * x + N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

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

Error

Derivation

  1. Initial program 7.2

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

  2. Simplified0.1

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

  3. Taylor expanded in x around 0 0.1

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022192 
(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))