?

Average Error: 7.3 → 0.1
Time: 19.2s
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
Error0.1
Cost13696
\[\left(\left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) + \log y \cdot \left(x + -1\right)\right) - t \]
Alternative 2
Error0.7
Cost7616
\[\left(z \cdot y + \left(\left(\log y \cdot \left(x + -1\right) - z \cdot y\right) - z \cdot y\right)\right) - t \]
Alternative 3
Error1.7
Cost7497
\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+19} \lor \neg \left(x + -1 \leq -1\right):\\ \;\;\;\;\left(x \cdot \log y - z \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-y\right) - \log y\right) - t\\ \end{array} \]
Alternative 4
Error3.0
Cost7433
\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -1.0002 \lor \neg \left(x + -1 \leq -1\right):\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-y\right) - \log y\right) - t\\ \end{array} \]
Alternative 5
Error11.0
Cost7380
\[\begin{array}{l} t_1 := \left(-\log y\right) - t\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -0.00018:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-125}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 6
Error10.2
Cost7376
\[\begin{array}{l} t_1 := \log y \cdot \left(x + -1\right) - t\\ \mathbf{if}\;x \leq -6 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-125}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-73}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error17.5
Cost7248
\[\begin{array}{l} t_1 := \log y \cdot \left(x + -1\right)\\ \mathbf{if}\;x \leq -0.00016:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-123}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error17.6
Cost7120
\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -0.00027:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-123}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-73}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error0.7
Cost7104
\[\left(\log y \cdot \left(x + -1\right) - z \cdot y\right) - t \]
Alternative 10
Error21.7
Cost6857
\[\begin{array}{l} \mathbf{if}\;x \leq -0.00027 \lor \neg \left(x \leq 4.4 \cdot 10^{+77}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \]
Alternative 11
Error36.9
Cost520
\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 12
Error34.3
Cost448
\[y \cdot \left(1 - z\right) - t \]
Alternative 13
Error34.4
Cost384
\[z \cdot \left(-y\right) - t \]
Alternative 14
Error41.3
Cost128
\[-t \]
Alternative 15
Error62.2
Cost64
\[y \]

Error

Reproduce?

herbie shell --seed 2023031 
(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))