Average Error: 0.0 → 0.0
Time: 6.0s
Precision: binary64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
↓
\[\begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ \frac{1 + {t_1}^{2}}{2 + \log \left(e^{t_1 \cdot t_1}\right)} \end{array} \]
\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\begin{array}{l}
t_1 := \frac{2 \cdot t}{1 + t}\\
\frac{1 + {t_1}^{2}}{2 + \log \left(e^{t_1 \cdot t_1}\right)}
\end{array}

(FPCore (t)
 :precision binary64
 (/
  (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))
  (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))
↓
(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))))
   (/ (+ 1.0 (pow t_1 2.0)) (+ 2.0 (log (exp (* t_1 t_1)))))))
double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}
↓
double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	return (1.0 + pow(t_1, 2.0)) / (2.0 + log(exp(t_1 * t_1)));
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

  2. Applied add-log-exp_binary640.0

    \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \color{blue}{\log \left(e^{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\right)}} \]

  3. Applied pow2_binary640.0

    \[\leadsto \frac{1 + \color{blue}{{\left(\frac{2 \cdot t}{1 + t}\right)}^{2}}}{2 + \log \left(e^{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\right)} \]

  4. Final simplification0.0

    \[\leadsto \frac{1 + {\left(\frac{2 \cdot t}{1 + t}\right)}^{2}}{2 + \log \left(e^{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\right)} \]

Reproduce

herbie shell --seed 2022024 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))