Average Error: 0.1 → 0.0
Time: 954.0ms
Precision: binary64
\[\frac{x + y}{y + y} \]
↓
\[\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right) \]
(FPCore (x y) :precision binary64 (/ (+ x y) (+ y y)))
↓
(FPCore (x y) :precision binary64 (fma 0.5 (/ x y) 0.5))
double code(double x, double y) {
	return (x + y) / (y + y);
}
↓
double code(double x, double y) {
	return fma(0.5, (x / y), 0.5);
}

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + y))
end
↓
function code(x, y)
	return fma(0.5, Float64(x / y), 0.5)
end

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(0.5 * N[(x / y), $MachinePrecision] + 0.5), $MachinePrecision]

\frac{x + y}{y + y}
↓
\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.0
Herbie0.0
\[0.5 \cdot \frac{x}{y} + 0.5 \]

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{x}{y} + 0.5} \]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)} \]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"
  :precision binary64

  :herbie-target
  (+ (* 0.5 (/ x y)) 0.5)

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