Average Error: 0.0 → 0.0
Time: 1.5s
Precision: binary64
\[\frac{1}{x - 1} + \frac{x}{x + 1} \]
↓
\[\frac{1 + x}{\mathsf{fma}\left(x, x, -1\right)} + \frac{x}{1 + x} \]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{1 + x}{\mathsf{fma}\left(x, x, -1\right)} + \frac{x}{1 + x}

(FPCore (x) :precision binary64 (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (+ (/ (+ 1.0 x) (fma x x -1.0)) (/ x (+ 1.0 x))))
double code(double x) {
	return (1.0 / (x - 1.0)) + (x / (x + 1.0));
}
↓
double code(double x) {
	return ((1.0 + x) / fma(x, x, -1.0)) + (x / (1.0 + x));
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Applied flip--_binary640.0

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}} + \frac{x}{x + 1} \]

  3. Applied associate-/r/_binary640.0

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

  4. Simplified0.0

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

  5. Applied associate-*l/_binary640.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022020 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))