Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[\frac{x - y}{x + y} \]
↓
\[\mathsf{fma}\left(1, \frac{x}{x + y}, \frac{-y}{x + y}\right) \]
(FPCore (x y) :precision binary64 (/ (- x y) (+ x y)))
↓
(FPCore (x y) :precision binary64 (fma 1.0 (/ x (+ x y)) (/ (- y) (+ x y))))
double code(double x, double y) {
	return (x - y) / (x + y);
}
↓
double code(double x, double y) {
	return fma(1.0, (x / (x + y)), (-y / (x + y)));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\frac{x}{x + y} - \frac{y}{x + y} \]

Derivation

  1. Initial program 0.0

    \[\frac{x - y}{x + y} \]

  2. Applied egg-rr0.8

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{x + y}\right)\right)} \]

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

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

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