cos2 (problem 3.4.1)

Average Error: 31.2 → 0.3

Time: 17.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.030963531967900033:\\ \;\;\;\;\frac{1}{x} \cdot \frac{e^{\log \left(1 - \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.029848537881546774:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

C

double f(double x) {
        double r18548 = 1.0;
        double r18549 = x;
        double r18550 = cos(r18549);
        double r18551 = r18548 - r18550;
        double r18552 = r18549 * r18549;
        double r18553 = r18551 / r18552;
        return r18553;
}

↓

double f(double x) {
        double r18554 = x;
        double r18555 = -0.030963531967900033;
        bool r18556 = r18554 <= r18555;
        double r18557 = 1.0;
        double r18558 = r18557 / r18554;
        double r18559 = 1.0;
        double r18560 = cos(r18554);
        double r18561 = r18559 - r18560;
        double r18562 = log(r18561);
        double r18563 = exp(r18562);
        double r18564 = r18563 / r18554;
        double r18565 = r18558 * r18564;
        double r18566 = 0.029848537881546774;
        bool r18567 = r18554 <= r18566;
        double r18568 = 0.001388888888888889;
        double r18569 = 4.0;
        double r18570 = pow(r18554, r18569);
        double r18571 = r18568 * r18570;
        double r18572 = 0.5;
        double r18573 = r18571 + r18572;
        double r18574 = 0.041666666666666664;
        double r18575 = 2.0;
        double r18576 = pow(r18554, r18575);
        double r18577 = r18574 * r18576;
        double r18578 = r18573 - r18577;
        double r18579 = r18561 / r18554;
        double r18580 = r18579 / r18554;
        double r18581 = r18567 ? r18578 : r18580;
        double r18582 = r18556 ? r18565 : r18581;
        return r18582;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.030963531967900033

   1. Initial program 1.1

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

   3. Applied associate-/r* 0.5

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

   5. Applied add-sqr-sqrt 64.0

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

   6. Applied add-sqr-sqrt 64.0

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

   7. Applied *-un-lft-identity 64.0

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

   8. Applied times-frac 64.0

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

   9. Applied times-frac 64.0

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

   10. Simplified 64.0

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

   11. Simplified 0.5

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

   13. Applied add-exp-log 0.5

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

   if -0.030963531967900033 < x < 0.029848537881546774

   1. Initial program 62.4

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

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

   if 0.029848537881546774 < x

   1. Initial program 0.8

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

   3. Applied associate-/r* 0.5

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

   5. Applied add-sqr-sqrt 0.6

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

   6. Applied add-sqr-sqrt 0.7

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

   7. Applied *-un-lft-identity 0.7

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

   8. Applied times-frac 0.7

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

   9. Applied times-frac 0.7

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

   10. Simplified 0.6

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

   11. Simplified 0.5

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

   13. Applied *-un-lft-identity 0.5

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

   14. Applied *-un-lft-identity 0.5

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

   15. Applied times-frac 0.5

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

   16. Applied associate-*l* 0.5

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

   17. Simplified 0.5

       \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030963531967900033:\\ \;\;\;\;\frac{1}{x} \cdot \frac{e^{\log \left(1 - \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.029848537881546774:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))