?

Average Error: 7.3 → 0.1
Time: 18.2s
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 7.3

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

  2. Simplified0.1

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

    [Start]7.3

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

    +-commutative [=>]7.3

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

    fma-def [=>]7.3

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

    sub-neg [=>]7.3

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

    log1p-def [=>]0.1

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

    remove-double-neg [<=]0.1

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

    remove-double-neg [=>]0.1

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

    sub-neg [=>]0.1

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

    metadata-eval [=>]0.1

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

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error0.3
Cost7616
\[\left(\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\right) - t \]
Alternative 2
Error3.5
Cost7496
\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]
Alternative 3
Error10.0
Cost7380
\[\begin{array}{l} t_1 := x \cdot \log y - t\\ t_2 := \left(-\log y\right) - t\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -650000000:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error0.6
Cost7360
\[\left(\frac{\log y}{\frac{1}{x + -1}} + y \cdot \left(1 - z\right)\right) - t \]
Alternative 5
Error7.0
Cost7245
\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+248} \lor \neg \left(z \leq 4.5 \cdot 10^{+226}\right) \land z \leq 3.5 \cdot 10^{+265}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]
Alternative 6
Error0.5
Cost7232
\[\left(\log y \cdot \left(x + -1\right) + y \cdot \left(1 - z\right)\right) - t \]
Alternative 7
Error9.2
Cost7116
\[\begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -9 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1050000000:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-10}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error0.6
Cost7104
\[\left(\log y \cdot \left(x + -1\right) - z \cdot y\right) - t \]
Alternative 9
Error15.4
Cost7052
\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -700000000:\\ \;\;\;\;\left(\left(z + -1\right) \cdot y\right) \cdot \left(-1 + y \cdot -0.5\right) - t\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+24}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error21.3
Cost6857
\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+63} \lor \neg \left(x \leq 1.16 \cdot 10^{+24}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + -1\right) \cdot y\right) \cdot \left(-1 + y \cdot -0.5\right) - t\\ \end{array} \]
Alternative 11
Error34.2
Cost832
\[\left(\left(z + -1\right) \cdot y\right) \cdot \left(-1 + y \cdot -0.5\right) - t \]
Alternative 12
Error34.4
Cost704
\[z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t \]
Alternative 13
Error36.7
Cost520
\[\begin{array}{l} \mathbf{if}\;t \leq -800000000000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 40000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 14
Error34.4
Cost448
\[y \cdot \left(1 - z\right) - t \]
Alternative 15
Error34.5
Cost384
\[z \cdot \left(-y\right) - t \]
Alternative 16
Error41.5
Cost128
\[-t \]
Alternative 17
Error62.5
Cost64
\[t \]

Error

Reproduce?

herbie shell --seed 2023018 
(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))