sintan (problem 3.4.5)

Average Error: 31.3 → 0.0

Time: 11.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.026896790136361697:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x + \left(-\tan x\right)}}\right)\\ \mathbf{elif}\;x \le 0.027208711435501284:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x + \left(-\tan x\right)}\right)}^{3}}\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((x <= -0.026896790136361697)) {
		temp = log(exp(((x - sin(x)) / (x + -tan(x)))));
	} else {
		double temp_1;
		if ((x <= 0.027208711435501284)) {
			temp_1 = fma(0.225, pow(x, 2.0), -fma(0.009642857142857142, pow(x, 4.0), 0.5));
		} else {
			temp_1 = cbrt(pow(((x - sin(x)) / (x + -tan(x))), 3.0));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.026896790136361697

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 1.4

      \[\leadsto \frac{x - \sin x}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \tan x}\]

   4. Applied fma-neg 1.4

      \[\leadsto \frac{x - \sin x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 41.3

      \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}}\]

   7. Applied add-cbrt-cube 42.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}\]

   8. Applied cbrt-undiv 42.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}}\]

   9. Simplified 1.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}\right)}^{3}}}\]
   10. Using strategy rm

   11. Applied fma-udef 1.1

       \[\leadsto \sqrt[3]{{\left(\frac{x - \sin x}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x} + \left(-\tan x\right)}}\right)}^{3}}\]

   12. Simplified 0.1

       \[\leadsto \sqrt[3]{{\left(\frac{x - \sin x}{\color{blue}{x} + \left(-\tan x\right)}\right)}^{3}}\]
   13. Using strategy rm

   14. Applied add-log-exp 0.1

       \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{{\left(\frac{x - \sin x}{x + \left(-\tan x\right)}\right)}^{3}}}\right)}\]

   15. Simplified 0.1

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

   if -0.026896790136361697 < x < 0.027208711435501284

   1. Initial program 63.0

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

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

   if 0.027208711435501284 < x

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 1.4

      \[\leadsto \frac{x - \sin x}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \tan x}\]

   4. Applied fma-neg 1.4

      \[\leadsto \frac{x - \sin x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 41.2

      \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}}\]

   7. Applied add-cbrt-cube 42.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}\]

   8. Applied cbrt-undiv 42.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}}}\]

   9. Simplified 1.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\tan x\right)}\right)}^{3}}}\]
   10. Using strategy rm

   11. Applied fma-udef 1.1

       \[\leadsto \sqrt[3]{{\left(\frac{x - \sin x}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x} + \left(-\tan x\right)}}\right)}^{3}}\]

   12. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026896790136361697:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x + \left(-\tan x\right)}}\right)\\ \mathbf{elif}\;x \le 0.027208711435501284:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x + \left(-\tan x\right)}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))