Average Error: 0.5 → 0.6
Time: 24.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - \left(\left(v \cdot v\right) \cdot 5\right) \cdot \left(\left(v \cdot v\right) \cdot 5\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + \left(v \cdot v\right) \cdot 5\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - \left(\left(v \cdot v\right) \cdot 5\right) \cdot \left(\left(v \cdot v\right) \cdot 5\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + \left(v \cdot v\right) \cdot 5\right)}\right)
double f(double v) {
        double r6655291 = 1.0;
        double r6655292 = 5.0;
        double r6655293 = v;
        double r6655294 = r6655293 * r6655293;
        double r6655295 = r6655292 * r6655294;
        double r6655296 = r6655291 - r6655295;
        double r6655297 = r6655294 - r6655291;
        double r6655298 = r6655296 / r6655297;
        double r6655299 = acos(r6655298);
        return r6655299;
}
↓
double f(double v) {
        double r6655300 = atan2(1.0, 0.0);
        double r6655301 = 2.0;
        double r6655302 = r6655300 / r6655301;
        double r6655303 = 1.0;
        double r6655304 = v;
        double r6655305 = r6655304 * r6655304;
        double r6655306 = 5.0;
        double r6655307 = r6655305 * r6655306;
        double r6655308 = r6655307 * r6655307;
        double r6655309 = r6655303 - r6655308;
        double r6655310 = r6655305 - r6655303;
        double r6655311 = r6655303 + r6655307;
        double r6655312 = r6655310 * r6655311;
        double r6655313 = r6655309 / r6655312;
        double r6655314 = asin(r6655313);
        double r6655315 = r6655302 - r6655314;
        return r6655315;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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--0.6

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

  4. Applied associate-/l/0.6

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

  6. Applied acos-asin0.6

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)}\]

  7. Final simplification0.6

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{1 - \left(\left(v \cdot v\right) \cdot 5\right) \cdot \left(\left(v \cdot v\right) \cdot 5\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + \left(v \cdot v\right) \cdot 5\right)}\right)\]

Reproduce

herbie shell --seed 2019141 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))