tanhf (example 3.4)

Average Error: 30.8 → 0.6

Time: 8.9s

Precision: binary64

Math

\[\frac{1 - \cos x}{\sin x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.017330074816333453:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x} \cdot \frac{1}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}\\ \mathbf{elif}\;x \leq 0.02149337157003914:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}}}\right)\\ \end{array}\]

C

double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -0.017330074816333453)) {
		VAR = ((double) ((((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) sin(x))) * (1.0 / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.02149337157003914)) {
			VAR_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
		} else {
			VAR_1 = ((double) (((double) log(((double) sqrt(((double) exp((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))))))))) + ((double) log(((double) sqrt(((double) exp((((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))) / ((double) sin(x)))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.8
Target	0.0
Herbie	0.6

\[\tan \left(\frac{x}{2}\right)\]

Derivation

1. Split input into 3 regimes

2. if x < -0.017330074816333453

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied clear-num 0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
   4. Using strategy rm

   5. Applied flip3-- 1.0

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}}\]

   6. Applied associate-/r/ 1.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

   7. Applied add-sqr-sqrt 1.0

      \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{\sin x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}\]

   8. Applied times-frac 1.1

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\frac{\sin x}{{1}^{3} - {\left(\cos x\right)}^{3}}} \cdot \frac{\sqrt{1}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\]

   9. Simplified 1.0

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}} \cdot \frac{\sqrt{1}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\]

   10. Simplified 1.0

       \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x} \cdot \color{blue}{\frac{1}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}\]

   if -0.017330074816333453 < x < 0.0214933715700391416

   1. Initial program 59.9

      \[\frac{1 - \cos x}{\sin x}\]

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)}\]

   if 0.0214933715700391416 < x

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-log-exp 1.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 1.2

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}} \cdot \sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)}\]

   6. Applied log-prod 1.2

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)}\]
   7. Using strategy rm

   8. Applied add-log-exp 1.2

      \[\leadsto \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{1 - \color{blue}{\log \left(e^{\cos x}\right)}}{\sin x}}}\right)\]

   9. Applied add-log-exp 1.2

      \[\leadsto \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}{\sin x}}}\right)\]

   10. Applied diff-log 1.3

       \[\leadsto \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{\sin x}}}\right)\]

   11. Simplified 1.2

       \[\leadsto \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{\log \color{blue}{\left(e^{1 - \cos x}\right)}}{\sin x}}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.017330074816333453:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x} \cdot \frac{1}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}\\ \mathbf{elif}\;x \leq 0.02149337157003914:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))