Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.5

Time: 9.1s

Precision: 64

Math

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

↓

\[\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\]

C

double f(double v) {
        double r232441 = 1.0;
        double r232442 = 5.0;
        double r232443 = v;
        double r232444 = r232443 * r232443;
        double r232445 = r232442 * r232444;
        double r232446 = r232441 - r232445;
        double r232447 = r232444 - r232441;
        double r232448 = r232446 / r232447;
        double r232449 = acos(r232448);
        return r232449;
}

↓

double f(double v) {
        double r232450 = 1.0;
        double r232451 = r232450 * r232450;
        double r232452 = 5.0;
        double r232453 = v;
        double r232454 = r232453 * r232453;
        double r232455 = r232452 * r232454;
        double r232456 = r232455 * r232455;
        double r232457 = r232451 - r232456;
        double r232458 = r232454 - r232450;
        double r232459 = r232450 + r232455;
        double r232460 = r232458 * r232459;
        double r232461 = r232457 / r232460;
        double r232462 = acos(r232461);
        return r232462;
}

Error

Bits error versus v

Derivation

1. Initial program 0.5

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
2. Using strategy rm

3. Applied flip-- 0.5

   \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{1 + 5 \cdot \left(v \cdot v\right)}}}{v \cdot v - 1}\right)\]

4. Applied associate-/l/ 0.5

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

6. Applied add-cbrt-cube 1.5

   \[\leadsto \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right) \cdot \cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\right) \cdot \cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)}}\]

7. Simplified 1.5

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

9. Applied pow1/3 0.5

   \[\leadsto \color{blue}{{\left({\left(\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\right)}^{3}\right)}^{\frac{1}{3}}}\]

10. Final simplification 0.5

    \[\leadsto \cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019308 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))