Average Error: 31.6 → 0.0
Time: 10.4s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.028750709088636438 \lor \neg \left(x \leq 0.030914659670639447\right):\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.009642857142857142, -0.5\right)\right)\\ \end{array} \]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.028750709088636438 \lor \neg \left(x \leq 0.030914659670639447\right):\\
\;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.009642857142857142, -0.5\right)\right)\\


\end{array}

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.028750709088636438) (not (<= x 0.030914659670639447)))
   (/ (- x (sin x)) (- x (/ (sin x) (cos x))))
   (fma 0.225 (* x x) (fma (pow x 4.0) -0.009642857142857142 -0.5))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.028750709088636438) || !(x <= 0.030914659670639447)) {
		tmp = (x - sin(x)) / (x - (sin(x) / cos(x)));
	} else {
		tmp = fma(0.225, (x * x), fma(pow(x, 4.0), -0.009642857142857142, -0.5));
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.028750709088636438 or 0.0309146596706394473 < x

    1. Initial program 0.0

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

    2. Taylor expanded in x around inf 0.0

      \[\leadsto \frac{x - \sin x}{\color{blue}{x - \frac{\sin x}{\cos x}}} \]

    if -0.028750709088636438 < x < 0.0309146596706394473

    1. Initial program 63.0

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

    2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)} \]

    3. Simplified0.0

      \[\leadsto \color{blue}{0.225 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(0.009642857142857142, {x}^{4}, 0.5\right)} \]

    4. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)} \]

    5. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.009642857142857142, -0.5\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.028750709088636438 \lor \neg \left(x \leq 0.030914659670639447\right):\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left({x}^{4}, -0.009642857142857142, -0.5\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))