Average Error: 31.4 → 0.5
Time: 13.1s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.619158073142093:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.004331643272607194:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.619158073142093:\\
\;\;\;\;1\\

\mathbf{elif}\;x \leq 0.004331643272607194:\\
\;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2.619158073142093)
   1.0
   (if (<= x 0.004331643272607194)
     (+ -0.5 (* 0.225 (* x x)))
     (/ 1.0 (/ (- x (tan x)) (- x (sin x)))))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -2.619158073142093) {
		tmp = 1.0;
	} else if (x <= 0.004331643272607194) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = 1.0 / ((x - tan(x)) / (x - sin(x)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -2.6191580731420929

    1. Initial program 0.0

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

    2. Taylor expanded around inf 1.6

      \[\leadsto \color{blue}{1}\]

    if -2.6191580731420929 < x < 0.00433164327260719424

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.00433164327260719424 < x

    1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]
    2. Using strategy rm

    3. Applied clear-num_binary640.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.619158073142093:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.004331643272607194:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))