tanhf (example 3.4)

Average Error: 30.7 → 1.1

Time: 24.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.04589551321338635608837464019416074734181:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \cos x + 1 \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1}\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.04166666666666667129259593593815225176513 + 0.5\right) \cdot x + {x}^{5} \cdot 0.004166666666666666608842550800773096852936\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \frac{\sqrt[3]{\left(\left(\left(\cos x + \cos \left(x + x\right) \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos x + \cos \left(x + x\right) \cdot \cos x\right) \cdot \cos x\right)\right) \cdot \cos x}}{\sqrt[3]{4}}}{\sin x \cdot \left(1 \cdot \left(1 + \cos x\right) + \cos x \cdot \cos x\right)}\\ \end{array}\]

C

double f(double x) {
        double r3347544 = 1.0;
        double r3347545 = x;
        double r3347546 = cos(r3347545);
        double r3347547 = r3347544 - r3347546;
        double r3347548 = sin(r3347545);
        double r3347549 = r3347547 / r3347548;
        return r3347549;
}

↓

double f(double x) {
        double r3347550 = 1.0;
        double r3347551 = x;
        double r3347552 = cos(r3347551);
        double r3347553 = r3347550 - r3347552;
        double r3347554 = sin(r3347551);
        double r3347555 = r3347553 / r3347554;
        double r3347556 = -0.045895513213386356;
        bool r3347557 = r3347555 <= r3347556;
        double r3347558 = 3.0;
        double r3347559 = pow(r3347550, r3347558);
        double r3347560 = pow(r3347552, r3347558);
        double r3347561 = r3347559 - r3347560;
        double r3347562 = r3347552 * r3347552;
        double r3347563 = r3347550 * r3347550;
        double r3347564 = r3347562 - r3347563;
        double r3347565 = r3347552 - r3347550;
        double r3347566 = r3347564 / r3347565;
        double r3347567 = r3347550 * r3347566;
        double r3347568 = r3347562 + r3347567;
        double r3347569 = r3347554 * r3347568;
        double r3347570 = r3347561 / r3347569;
        double r3347571 = -0.0;
        bool r3347572 = r3347555 <= r3347571;
        double r3347573 = r3347551 * r3347551;
        double r3347574 = 0.04166666666666667;
        double r3347575 = r3347573 * r3347574;
        double r3347576 = 0.5;
        double r3347577 = r3347575 + r3347576;
        double r3347578 = r3347577 * r3347551;
        double r3347579 = 5.0;
        double r3347580 = pow(r3347551, r3347579);
        double r3347581 = 0.004166666666666667;
        double r3347582 = r3347580 * r3347581;
        double r3347583 = r3347578 + r3347582;
        double r3347584 = r3347551 + r3347551;
        double r3347585 = cos(r3347584);
        double r3347586 = r3347585 * r3347552;
        double r3347587 = r3347552 + r3347586;
        double r3347588 = r3347587 * r3347552;
        double r3347589 = r3347588 * r3347588;
        double r3347590 = r3347589 * r3347552;
        double r3347591 = cbrt(r3347590);
        double r3347592 = 4.0;
        double r3347593 = cbrt(r3347592);
        double r3347594 = r3347591 / r3347593;
        double r3347595 = r3347559 - r3347594;
        double r3347596 = r3347550 + r3347552;
        double r3347597 = r3347550 * r3347596;
        double r3347598 = r3347597 + r3347562;
        double r3347599 = r3347554 * r3347598;
        double r3347600 = r3347595 / r3347599;
        double r3347601 = r3347572 ? r3347583 : r3347600;
        double r3347602 = r3347557 ? r3347570 : r3347601;
        return r3347602;
}

Error

Bits error versus x

Target

Original	30.7
Target	0.0
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.045895513213386356

   1. Initial program 0.7

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

   3. Applied flip3-- 0.8

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

   4. Applied associate-/l/ 0.9

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

   5. Simplified 0.9

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

   7. Applied flip-+ 0.9

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

   if -0.045895513213386356 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

   1. Initial program 59.6

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

   3. Applied div-sub 59.6

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]

   4. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.04166666666666667129259593593815225176513 \cdot {x}^{3} + 0.004166666666666666608842550800773096852936 \cdot {x}^{5}\right)}\]

   5. Simplified 0.9

      \[\leadsto \color{blue}{{x}^{5} \cdot 0.004166666666666666608842550800773096852936 + x \cdot \left(\left(x \cdot x\right) \cdot 0.04166666666666667129259593593815225176513 + 0.5\right)}\]

   if -0.0 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.6

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

   3. Applied flip3-- 1.7

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

   4. Applied associate-/l/ 1.7

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

   5. Simplified 1.7

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

   7. Applied add-cbrt-cube 1.8

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

   8. Simplified 1.8

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

   10. Applied cos-mult 1.8

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

   11. Applied associate-*l/ 1.8

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

   12. Applied cos-mult 1.8

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

   13. Applied associate-*l/ 1.8

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

   14. Applied associate-*r/ 1.8

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

   15. Applied frac-times 1.8

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

   16. Applied cbrt-div 1.8

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

   17. Simplified 1.8

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

   18. Simplified 1.8

       \[\leadsto \frac{{1}^{3} - \frac{\sqrt[3]{\left(\left(\left(\cos x \cdot \cos \left(x + x\right) + \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos x \cdot \cos \left(x + x\right) + \cos x\right) \cdot \cos x\right)\right) \cdot \cos x}}{\color{blue}{\sqrt[3]{4}}}}{\left(\cos x \cdot \cos x + 1 \cdot \left(\cos x + 1\right)\right) \cdot \sin x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.04589551321338635608837464019416074734181:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \cos x + 1 \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1}\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.04166666666666667129259593593815225176513 + 0.5\right) \cdot x + {x}^{5} \cdot 0.004166666666666666608842550800773096852936\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \frac{\sqrt[3]{\left(\left(\left(\cos x + \cos \left(x + x\right) \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos x + \cos \left(x + x\right) \cdot \cos x\right) \cdot \cos x\right)\right) \cdot \cos x}}{\sqrt[3]{4}}}{\sin x \cdot \left(1 \cdot \left(1 + \cos x\right) + \cos x \cdot \cos x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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