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

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))) ^ -1.0
end

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

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

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\frac{1}{\frac{x}{x + y} - \frac{y}{x + y}} \]

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022153 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

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