?

Average Accuracy: 89.5% → 99.8%
Time: 16.9s
Precision: binary64
Cost: 19968

?

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

Error?

Derivation?

  1. Initial program 89.5%

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

  2. Simplified99.8%

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

    [Start]89.5

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

    fma-def [=>]89.5

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

    sub-neg [=>]89.5

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

    log1p-def [=>]99.8

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

  3. Final simplification99.8%

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

Alternatives

Alternative 1
Accuracy99.7%
Cost13696
\[z \cdot \mathsf{log1p}\left(-y\right) + \left(\left(y + \left(x + -1\right) \cdot \log y\right) - t\right) \]
Alternative 2
Accuracy99.3%
Cost7744
\[z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) + \left(\left(y + \frac{\log y}{\frac{1}{x + -1}}\right) - t\right) \]
Alternative 3
Accuracy99.4%
Cost7616
\[\left(\left(y + \left(x + -1\right) \cdot \log y\right) - t\right) + z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) \]
Alternative 4
Accuracy95.2%
Cost7496
\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.999999998:\\ \;\;\;\;\left(\left(y - y \cdot z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]
Alternative 5
Accuracy99.1%
Cost7232
\[\left(\left(x + -1\right) \cdot \log y + y \cdot \left(1 - z\right)\right) - t \]
Alternative 6
Accuracy99.1%
Cost7232
\[\left(y + \left(\left(x + -1\right) \cdot \log y - y \cdot z\right)\right) - t \]
Alternative 7
Accuracy87.5%
Cost6985
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
Alternative 8
Accuracy87.6%
Cost6985
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
Alternative 9
Accuracy89.7%
Cost6980
\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+244}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]
Alternative 10
Accuracy61.4%
Cost6921
\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+116} \lor \neg \left(z \leq 3 \cdot 10^{+101}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
Alternative 11
Accuracy56.3%
Cost6793
\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-48} \lor \neg \left(t \leq 7.5 \cdot 10^{-15}\right):\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \]
Alternative 12
Accuracy47.2%
Cost704
\[z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t \]
Alternative 13
Accuracy43.1%
Cost520
\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+18}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 14
Accuracy47.1%
Cost448
\[\left(y - y \cdot z\right) - t \]
Alternative 15
Accuracy46.9%
Cost384
\[\left(-t\right) - y \cdot z \]
Alternative 16
Accuracy36.9%
Cost128
\[-t \]

Error

Reproduce?

herbie shell --seed 2023138 
(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))