?

Average Error: 10.39% → 0.17%
Time: 17.8s
Precision: binary64
Cost: 19968

?

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
↓
\[\mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \log y \cdot \left(x + -1\right)\right) - t \]
(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 (+ z -1.0) (log1p (- y)) (* (log y) (+ x -1.0))) 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((z + -1.0), log1p(-y), (log(y) * (x + -1.0))) - 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(Float64(z + -1.0), log1p(Float64(-y)), Float64(log(y) * Float64(x + -1.0))) - 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[(N[(z + -1.0), $MachinePrecision] * N[Log[1 + (-y)], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 10.39

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

  2. Simplified0.17

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

    [Start]10.39

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

    +-commutative [=>]10.39

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

    fma-def [=>]10.39

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

    sub-neg [=>]10.39

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

    log1p-def [=>]0.17

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

    remove-double-neg [<=]0.17

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

    remove-double-neg [=>]0.17

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

    sub-neg [=>]0.17

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

    metadata-eval [=>]0.17

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

  3. Final simplification0.17

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

Alternatives

Alternative 1
Error0.22%
Cost13824
\[\left(\log y \cdot \left(x + -1\right) + \frac{\mathsf{log1p}\left(-y\right)}{\frac{1}{z + -1}}\right) - t \]
Alternative 2
Error0.54%
Cost7744
\[\left(\log \left(\frac{1}{y}\right) \cdot \left(1 - x\right) + \left(y \cdot \left(y \cdot -0.5\right) - y\right) \cdot \left(z + -1\right)\right) - t \]
Alternative 3
Error0.5%
Cost7616
\[\left(\log y \cdot \left(x + -1\right) + \left(y \cdot \left(y \cdot -0.5\right) - y\right) \cdot \left(z + -1\right)\right) - t \]
Alternative 4
Error4.68%
Cost7496
\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -50:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
Alternative 5
Error0.84%
Cost7232
\[\left(\log y \cdot \left(x + -1\right) + y \cdot \left(1 - z\right)\right) - t \]
Alternative 6
Error14.42%
Cost7116
\[\begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-35}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error14.43%
Cost7116
\[\begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-35}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error12.42%
Cost6985
\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \lor \neg \left(x \leq 1.35 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
Alternative 9
Error13.6%
Cost6980
\[\begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
Alternative 10
Error23.99%
Cost6921
\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+25} \lor \neg \left(x \leq 9 \cdot 10^{+51}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
Alternative 11
Error33.29%
Cost6857
\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+52} \lor \neg \left(x \leq 4.5 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \]
Alternative 12
Error44.94%
Cost6792
\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-21}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-57}:\\ \;\;\;\;-\log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]
Alternative 13
Error54.12%
Cost704
\[z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t \]
Alternative 14
Error54.41%
Cost384
\[z \cdot \left(-y\right) - t \]
Alternative 15
Error64.27%
Cost128
\[-t \]

Error

Reproduce?

herbie shell --seed 2023090 
(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))