Average Error: 31.6 → 0.1
Time: 11.8s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005143713995356559 \lor \neg \left(x \leq 0.004699430420038539\right):\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \end{array} \]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.005143713995356559 \lor \neg \left(x \leq 0.004699430420038539\right):\\
\;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\


\end{array}

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.005143713995356559) (not (<= x 0.004699430420038539)))
   (log (exp (/ (- x (sin x)) (- x (tan x)))))
   (fma 0.225 (* x x) -0.5)))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.005143713995356559) || !(x <= 0.004699430420038539)) {
		tmp = log(exp((x - sin(x)) / (x - tan(x))));
	} else {
		tmp = fma(0.225, (x * x), -0.5);
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0051437139953565587 or 0.00469943042003853877 < x

    1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x} \]

    2. Applied add-log-exp_binary640.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)} \]

    if -0.0051437139953565587 < x < 0.00469943042003853877

    1. Initial program 63.3

      \[\frac{x - \sin x}{x - \tan x} \]

    2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.005143713995356559 \lor \neg \left(x \leq 0.004699430420038539\right):\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))