Average Error: 32.1 → 0.0
Time: 10.0s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.029873394671195968 \lor \neg \left(x \leq 0.02727798752832581\right):\\ \;\;\;\;\frac{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} - \frac{{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.029873394671195968 \lor \neg \left(x \leq 0.02727798752832581\right):\\
\;\;\;\;\frac{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} - \frac{{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.029873394671195968) (not (<= x 0.02727798752832581)))
   (/
    (-
     (* (/ x (- x (tan x))) (/ x (- x (tan x))))
     (/ (pow (sin x) 2.0) (* (- x (tan x)) (- x (tan x)))))
    (+ (/ x (- x (tan x))) (/ (sin x) (- x (tan x)))))
   (- (* (* x x) 0.225) (+ (* 0.009642857142857142 (pow x 4.0)) 0.5))))
double code(double x) {
	return (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))));
}
↓
double code(double x) {
	double tmp;
	if (((x <= -0.029873394671195968) || !(x <= 0.02727798752832581))) {
		tmp = (((double) (((double) ((x / ((double) (x - ((double) tan(x))))) * (x / ((double) (x - ((double) tan(x))))))) - (((double) pow(((double) sin(x)), 2.0)) / ((double) (((double) (x - ((double) tan(x)))) * ((double) (x - ((double) tan(x))))))))) / ((double) ((x / ((double) (x - ((double) tan(x))))) + (((double) sin(x)) / ((double) (x - ((double) tan(x))))))));
	} else {
		tmp = ((double) (((double) (((double) (x * x)) * 0.225)) - ((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5))));
	}
	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 2 regimes

  2. if x < -0.0298733946711959679 or 0.02727798752832581 < x

    1. Initial program 0.0

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

    3. Applied div-sub_binary640.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
    4. Using strategy rm

    5. Applied flip--_binary640.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}}\]

    6. Simplified0.1

      \[\leadsto \frac{\color{blue}{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} - \frac{{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\]

    if -0.0298733946711959679 < x < 0.02727798752832581

    1. Initial program 63.2

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029873394671195968 \lor \neg \left(x \leq 0.02727798752832581\right):\\ \;\;\;\;\frac{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} - \frac{{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))