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

↓

\[\mathsf{fma}\left(x, \log y, \left(-z\right) - y\right) \]

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

↓

(FPCore (x y z) :precision binary64 (fma x (log y) (- (- z) y)))

double code(double x, double y, double z) {
	return ((x * log(y)) - z) - y;
}

↓

double code(double x, double y, double z) {
	return fma(x, log(y), (-z - y));
}

function code(x, y, z)
	return Float64(Float64(Float64(x * log(y)) - z) - y)
end

↓

function code(x, y, z)
	return fma(x, log(y), Float64(Float64(-z) - y))
end

code[x_, y_, z_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]

↓

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

\left(x \cdot \log y - z\right) - y

↓

\mathsf{fma}\left(x, \log y, \left(-z\right) - y\right)

Derivation

Initial program 0.1
\[\left(x \cdot \log y - z\right) - y \]
Simplified0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)} \]
Proof
(fma.f64 x (log.f64 y) (neg.f64 (+.f64 y z))): 0 points increase in error, 0 points decrease in error
(fma.f64 x (log.f64 y) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 z y)))): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (log.f64 y)) (+.f64 z y))): 2 points increase in error, 1 points decrease in error
(Rewrite<= associate--l-_binary64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) z) y)): 0 points increase in error, 0 points decrease in error
Final simplification0.1
\[\leadsto \mathsf{fma}\left(x, \log y, \left(-z\right) - y\right) \]

Alternatives

Alternative 1
Error	9.4
Cost	7116

\[\begin{array}{l} t_0 := x \cdot \log y\\ t_1 := t_0 - y\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+14}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \]

Alternative 2
Error	9.3
Cost	6984

\[\begin{array}{l} t_0 := x \cdot \log y - y\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.7
Cost	6856

\[\begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+142}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.1
Cost	6848

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

Alternative 5
Error	30.8
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -1450000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	21.2
Cost	256

\[\left(-z\right) - y \]

Alternative 7
Error	42.0
Cost	128

\[-y \]

Alternative 8
Error	62.5
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))

Error

Derivation

Alternatives

Error

Reproduce