Average Error: 0.0 → 0.0
Time: 5.5s
Precision: binary64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{2}{1 + t}\\ \frac{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 5}{\mathsf{fma}\left(t_1, t_1 + -4, 6\right)} \end{array} \]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\begin{array}{l}
t_1 := \frac{2}{1 + t}\\
\frac{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 5}{\mathsf{fma}\left(t_1, t_1 + -4, 6\right)}
\end{array}

(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   2.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))
↓
(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (/
    (+ (/ (+ -8.0 (/ 4.0 (+ 1.0 t))) (+ 1.0 t)) 5.0)
    (fma t_1 (+ t_1 -4.0) 6.0))))
double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}
↓
double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return (((-8.0 + (4.0 / (1.0 + t))) / (1.0 + t)) + 5.0) / fma(t_1, (t_1 + -4.0), 6.0);
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)}} \]

  3. Applied fma-udef_binary640.0

    \[\leadsto \frac{\color{blue}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 5}}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)} \]

  4. Simplified0.0

    \[\leadsto \frac{\color{blue}{\frac{-8 + \frac{4}{1 + t}}{1 + t}} + 5}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)} \]

  5. Final simplification0.0

    \[\leadsto \frac{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 5}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)} \]

Reproduce

herbie shell --seed 2022024 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))