tanhf (example 3.4)

Average Error: 29.6 → 0.6

Time: 9.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.00595353611309251:\\ \;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0.00027129648320270975:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\\ \end{array}\]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

↓

(FPCore (x)
 :precision binary64
 (if (<= (/ (- 1.0 (cos x)) (sin x)) -0.00595353611309251)
   (/
    (- (pow 1.0 3.0) (cbrt (pow (pow (cos x) 3.0) 3.0)))
    (* (sin x) (+ (* 1.0 1.0) (* (cos x) (+ 1.0 (cos x))))))
   (if (<= (/ (- 1.0 (cos x)) (sin x)) 0.00027129648320270975)
     (+
      (* 0.041666666666666664 (pow x 3.0))
      (+ (* 0.004166666666666667 (pow x 5.0)) (* x 0.5)))
     (/
      (/
       (- (pow 1.0 3.0) (pow (cos x) 3.0))
       (+ (* 1.0 1.0) (* (cos x) (+ 1.0 (cos x)))))
      (sin x)))))

C

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

↓

double code(double x) {
	double VAR;
	if (((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= -0.00595353611309251)) {
		VAR = (((double) (((double) pow(1.0, 3.0)) - ((double) cbrt(((double) pow(((double) pow(((double) cos(x)), 3.0)), 3.0)))))) / ((double) (((double) sin(x)) * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))))));
	} else {
		double VAR_1;
		if (((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= 0.00027129648320270975)) {
			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) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x))))))))) / ((double) sin(x)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	29.6
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0059535361130925099

   1. Initial program 0.9

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

   3. Applied flip3-- 1.0

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

   4. Applied associate-/l/ 1.0

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

   5. Simplified 1.0

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

   7. Applied add-cbrt-cube 1.1

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

   8. Simplified 1.1

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

   if -0.0059535361130925099 < (/ (- 1.0 (cos x)) (sin x)) < 2.7129648320270975e-4

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

   if 2.7129648320270975e-4 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.0

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

   3. Applied flip3-- 1.1

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

   4. Simplified 1.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.00595353611309251:\\ \;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0.00027129648320270975:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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