Average Error: 32.1 → 0.0
Time: 10.8s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.029873394671195968 \lor \neg \left(x \leq 0.02727798752832581\right):\\ \;\;\;\;\frac{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} - \frac{{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + \left(-0.5 + {x}^{4} \cdot -0.009642857142857142\right)\\ \end{array}\]
\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}:\\
\;\;\;\;x \cdot \left(x \cdot 0.225\right) + \left(-0.5 + {x}^{4} \cdot -0.009642857142857142\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)) (+ -0.5 (* (pow x 4.0) -0.009642857142857142)))))
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) (x * ((double) (x * 0.225)))) + ((double) (-0.5 + ((double) (((double) pow(x, 4.0)) * -0.009642857142857142))))));
	}
	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 Error: 0.0 bits

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

    3. Applied div-subError: 0.0 bits

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

    5. Applied flip--Error: 0.1 bits

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

    6. SimplifiedError: 0.1 bits

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

    if -0.0298733946711959679 < x < 0.02727798752832581

    1. Initial program Error: 63.2 bits

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

    2. Taylor expanded around 0 Error: 0.0 bits

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

    3. SimplifiedError: 0.0 bits

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

  4. Final simplificationError: 0.0 bits

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

Reproduce

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