tanhf (example 3.4)

Average Error: 30.4 → 0.5

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.02393634628213539:\\ \;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}}}{\sin x}\\ \mathbf{elif}\;x \leq 0.0237498496172468:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}}}{\sin x}\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.02393634628213539)
   (/
    (/
     (- (pow 1.0 3.0) (log (exp (pow (cos x) 3.0))))
     (+
      (* 1.0 1.0)
      (* (cos x) (/ (- (* 1.0 1.0) (pow (cos x) 2.0)) (- 1.0 (cos x))))))
    (sin x))
   (if (<= x 0.0237498496172468)
     (+
      (* 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 1.0) (pow (cos x) 2.0)) (- 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 tmp;
	if ((x <= -0.02393634628213539)) {
		tmp = ((((double) (((double) pow(1.0, 3.0)) - ((double) log(((double) exp(((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * (((double) (((double) (1.0 * 1.0)) - ((double) pow(((double) cos(x)), 2.0)))) / ((double) (1.0 - ((double) cos(x)))))))))) / ((double) sin(x)));
	} else {
		double tmp_1;
		if ((x <= 0.0237498496172468)) {
			tmp_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
		} else {
			tmp_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) (((double) (1.0 * 1.0)) - ((double) pow(((double) cos(x)), 2.0)))) / ((double) (1.0 - ((double) cos(x)))))))))) / ((double) sin(x)));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	30.4
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if x < -0.023936346282135389

   1. Initial program Error: 0.9 bits

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

   3. Applied flip3-- Error: 1.0 bits

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

   4. Simplified Error: 1.0 bits

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

   6. Applied add-log-exp Error: 1.0 bits

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

   8. Applied flip-+ Error: 1.0 bits

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

   9. Simplified Error: 1.0 bits

      \[\leadsto \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \frac{\color{blue}{1 \cdot 1 - {\left(\cos x\right)}^{2}}}{1 - \cos x}}}{\sin x}\]

   if -0.023936346282135389 < x < 0.0237498496172468

   1. Initial program Error: 59.8 bits

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

   2. Taylor expanded around 0 Error: 0.0 bits

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

   3. Simplified Error: 0.0 bits

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

   if 0.0237498496172468 < x

   1. Initial program Error: 0.9 bits

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

   3. Applied flip3-- Error: 1.0 bits

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

   4. Simplified Error: 1.0 bits

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

   6. Applied flip-+ Error: 1.0 bits

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

   7. Simplified Error: 1.0 bits

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \frac{\color{blue}{1 \cdot 1 - {\left(\cos x\right)}^{2}}}{1 - \cos x}}}{\sin x}\]
3. Recombined 3 regimes into one program.

4. Final simplification Error: 0.5 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.02393634628213539:\\ \;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}}}{\sin x}\\ \mathbf{elif}\;x \leq 0.0237498496172468:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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