Average Error: 0.0 → 0.0
Time: 3.6s
Precision: binary64
\[\frac{x - y}{2 - \left(x + y\right)} \]
↓
\[\begin{array}{l} t_0 := 2 - \left(x + y\right)\\ \left(\frac{x}{t_0} - \frac{y}{t_0}\right) + \mathsf{fma}\left(\frac{-1}{t_0}, y, y \cdot \frac{1}{t_0}\right) \end{array} \]
(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 2.0 (+ x y))))
   (+ (- (/ x t_0) (/ y t_0)) (fma (/ -1.0 t_0) y (* y (/ 1.0 t_0))))))
double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}
↓
double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	return ((x / t_0) - (y / t_0)) + fma((-1.0 / t_0), y, (y * (1.0 / t_0)));
}

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end
↓
function code(x, y)
	t_0 = Float64(2.0 - Float64(x + y))
	return Float64(Float64(Float64(x / t_0) - Float64(y / t_0)) + fma(Float64(-1.0 / t_0), y, Float64(y * Float64(1.0 / t_0))))
end

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := Block[{t$95$0 = N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(x / t$95$0), $MachinePrecision] - N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / t$95$0), $MachinePrecision] * y + N[(y * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{x - y}{2 - \left(x + y\right)}
↓
\begin{array}{l}
t_0 := 2 - \left(x + y\right)\\
\left(\frac{x}{t_0} - \frac{y}{t_0}\right) + \mathsf{fma}\left(\frac{-1}{t_0}, y, y \cdot \frac{1}{t_0}\right)
\end{array}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\frac{x}{2 - \left(x + y\right)} - \frac{y}{2 - \left(x + y\right)} \]

Derivation

  1. Initial program 0.0

    \[\frac{x - y}{2 - \left(x + y\right)} \]

  2. Applied egg-rr0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2 - \left(x + y\right)}, -\frac{1}{2 - \left(x + y\right)} \cdot y\right) + \mathsf{fma}\left(-\frac{1}{2 - \left(x + y\right)}, y, \frac{1}{2 - \left(x + y\right)} \cdot y\right)} \]

  3. Applied egg-rr0.0

    \[\leadsto \color{blue}{\left(\left(\frac{x}{2 - \left(x + y\right)} + 0\right) - \frac{y}{2 - \left(x + y\right)}\right)} + \mathsf{fma}\left(-\frac{1}{2 - \left(x + y\right)}, y, \frac{1}{2 - \left(x + y\right)} \cdot y\right) \]

  4. Final simplification0.0

    \[\leadsto \left(\frac{x}{2 - \left(x + y\right)} - \frac{y}{2 - \left(x + y\right)}\right) + \mathsf{fma}\left(\frac{-1}{2 - \left(x + y\right)}, y, y \cdot \frac{1}{2 - \left(x + y\right)}\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, C"
  :precision binary64

  :herbie-target
  (- (/ x (- 2.0 (+ x y))) (/ y (- 2.0 (+ x y))))

  (/ (- x y) (- 2.0 (+ x y))))