sintan (problem 3.4.5)

Average Error: 31.7 → 0.0

Time: 11.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0292003739707403752:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{x - \tan x}, \frac{x}{x - \tan x}, \frac{-{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}\right)}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.024272639190728022:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \frac{x}{x - \tan x}, \frac{-{\left(\sin x\right)}^{2}}{x - \tan x}\right)}{x - \tan x}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \end{array}\]

C

double code(double x) {
	return ((x - sin(x)) / (x - tan(x)));
}

↓

double code(double x) {
	double temp;
	if ((x <= -0.029200373970740375)) {
		temp = (fma((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)))));
	} else {
		double temp_1;
		if ((x <= 0.024272639190728022)) {
			temp_1 = expm1(log1p(fma(0.225, pow(x, 2.0), -fma(0.009642857142857142, pow(x, 4.0), 0.5))));
		} else {
			temp_1 = ((fma(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)))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.029200373970740375

   1. Initial program 0.1

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

   3. Applied div-sub 0.1

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

   5. Applied flip-- 0.1

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

   7. Applied fma-neg 0.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{x - \tan x}, \frac{x}{x - \tan x}, -\frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}\right)}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\]

   8. Simplified 0.1

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

   if -0.029200373970740375 < x < 0.024272639190728022

   1. Initial program 63.2

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

   3. Applied expm1-log1p-u 63.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - \sin x}{x - \tan x}\right)\right)}\]

   4. Taylor expanded around 0 0.0

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)\right)\]

   5. Simplified 0.0

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)}\right)\right)\]

   if 0.024272639190728022 < x

   1. Initial program 0.0

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

   3. Applied div-sub 0.0

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

   5. Applied flip-- 0.0

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

   7. Applied associate-*r/ 0.0

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

   8. Applied associate-*r/ 0.1

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

   9. Applied sub-div 0.0

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

   10. Simplified 0.0

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x, \frac{x}{x - \tan x}, \frac{-{\left(\sin x\right)}^{2}}{x - \tan x}\right)}}{x - \tan x}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0292003739707403752:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{x - \tan x}, \frac{x}{x - \tan x}, \frac{-{\left(\sin x\right)}^{2}}{\left(x - \tan x\right) \cdot \left(x - \tan x\right)}\right)}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.024272639190728022:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \frac{x}{x - \tan x}, \frac{-{\left(\sin x\right)}^{2}}{x - \tan x}\right)}{x - \tan x}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))