arccos

Average Error: 0.0 → 0.0

Time: 14.9s

Precision: 64

Math

\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]

↓

\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{\left(1 \cdot 1\right)}^{3} - {\left(x \cdot x\right)}^{3}} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + 1 \cdot 1\right) + {1}^{4}\right) \cdot \left(1 - x\right)\right)}\right)\]

C

double f(double x) {
        double r23361 = 2.0;
        double r23362 = 1.0;
        double r23363 = x;
        double r23364 = r23362 - r23363;
        double r23365 = r23362 + r23363;
        double r23366 = r23364 / r23365;
        double r23367 = sqrt(r23366);
        double r23368 = atan(r23367);
        double r23369 = r23361 * r23368;
        return r23369;
}

↓

double f(double x) {
        double r23370 = 2.0;
        double r23371 = 1.0;
        double r23372 = x;
        double r23373 = r23371 - r23372;
        double r23374 = r23371 * r23371;
        double r23375 = 3.0;
        double r23376 = pow(r23374, r23375);
        double r23377 = r23372 * r23372;
        double r23378 = pow(r23377, r23375);
        double r23379 = r23376 - r23378;
        double r23380 = r23373 / r23379;
        double r23381 = r23377 + r23374;
        double r23382 = r23377 * r23381;
        double r23383 = 4.0;
        double r23384 = pow(r23371, r23383);
        double r23385 = r23382 + r23384;
        double r23386 = r23385 * r23373;
        double r23387 = r23380 * r23386;
        double r23388 = sqrt(r23387);
        double r23389 = atan(r23388);
        double r23390 = r23370 * r23389;
        return r23390;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
2. Using strategy rm

3. Applied flip-+ 0.0

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

4. Applied associate-/r/ 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{1 - x}{1 \cdot 1 - x \cdot x} \cdot \left(1 - x\right)}}\right)\]
5. Using strategy rm

6. Applied flip3-- 0.0

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

7. Applied associate-/r/ 0.0

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

8. Applied associate-*l* 0.0

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

9. Simplified 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{\left(1 \cdot 1\right)}^{3} - {\left(x \cdot x\right)}^{3}} \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + 1 \cdot 1\right) + {1}^{4}\right) \cdot \left(1 - x\right)\right)}}\right)\]

10. Final simplification 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{\left(1 \cdot 1\right)}^{3} - {\left(x \cdot x\right)}^{3}} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x + 1 \cdot 1\right) + {1}^{4}\right) \cdot \left(1 - x\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))