Average Error: 0.6 → 0.8
Time: 27.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\left(\left(v \cdot v\right) \cdot 4 - 1\right) + \left(\left(v \cdot v\right) \cdot 4\right) \cdot \left(v \cdot v\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\left(\left(v \cdot v\right) \cdot 4 - 1\right) + \left(\left(v \cdot v\right) \cdot 4\right) \cdot \left(v \cdot v\right)\right)
double f(double v) {
        double r4670275 = 1.0;
        double r4670276 = 5.0;
        double r4670277 = v;
        double r4670278 = r4670277 * r4670277;
        double r4670279 = r4670276 * r4670278;
        double r4670280 = r4670275 - r4670279;
        double r4670281 = r4670278 - r4670275;
        double r4670282 = r4670280 / r4670281;
        double r4670283 = acos(r4670282);
        return r4670283;
}
↓
double f(double v) {
        double r4670284 = atan2(1.0, 0.0);
        double r4670285 = 2.0;
        double r4670286 = r4670284 / r4670285;
        double r4670287 = v;
        double r4670288 = r4670287 * r4670287;
        double r4670289 = 4.0;
        double r4670290 = r4670288 * r4670289;
        double r4670291 = 1.0;
        double r4670292 = r4670290 - r4670291;
        double r4670293 = r4670290 * r4670288;
        double r4670294 = r4670292 + r4670293;
        double r4670295 = asin(r4670294);
        double r4670296 = r4670286 - r4670295;
        return r4670296;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around 0 0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]

  3. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(\left(v \cdot v\right) \cdot 4 - 1\right) + \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot 4\right)\right)}\]
  4. Using strategy rm

  5. Applied acos-asin0.8

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\left(\left(v \cdot v\right) \cdot 4 - 1\right) + \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot 4\right)\right)}\]

  6. Final simplification0.8

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\left(\left(v \cdot v\right) \cdot 4 - 1\right) + \left(\left(v \cdot v\right) \cdot 4\right) \cdot \left(v \cdot v\right)\right)\]

Reproduce

herbie shell --seed 2019146 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))