Average Error: 0.0 → 0.0

Time: 2.7s

Precision: binary64

Cost: 6976

\[0 \leq x \land x \leq 1\]

\[\frac{x}{x + 1} \]

↓

\[\frac{x}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + -1\right) \]

(FPCore (x) :precision binary64 (/ x (+ x 1.0)))

↓

(FPCore (x) :precision binary64 (* (/ x (fma x x -1.0)) (+ x -1.0)))

double code(double x) {
	return x / (x + 1.0);
}

↓

double code(double x) {
	return (x / fma(x, x, -1.0)) * (x + -1.0);
}

function code(x)
	return Float64(x / Float64(x + 1.0))
end

↓

function code(x)
	return Float64(Float64(x / fma(x, x, -1.0)) * Float64(x + -1.0))
end

code[x_] := N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(x / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]

\frac{x}{x + 1}

↓

\frac{x}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + -1\right)

Derivation

Initial program 0.0
\[\frac{x}{x + 1} \]
Applied egg-rr0.0
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + -1\right)} \]
Final simplification0.0
\[\leadsto \frac{x}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + -1\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	448

\[x \cdot \frac{1}{x + 1} \]

Alternative 2
Error	0.7
Cost	320

\[x - x \cdot x \]

Alternative 3
Error	0.0
Cost	320

\[\frac{x}{x + 1} \]

Alternative 4
Error	1.4
Cost	64

\[x \]

herbie shell --seed 1 
(FPCore (x)
  :name "x/(x+1)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1.0))
  (/ x (+ x 1.0)))