Average Error: 0.5 → 0.8
Time: 4.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)
double f(double v) {
        double r350168 = 1.0;
        double r350169 = 5.0;
        double r350170 = v;
        double r350171 = r350170 * r350170;
        double r350172 = r350169 * r350171;
        double r350173 = r350168 - r350172;
        double r350174 = r350171 - r350168;
        double r350175 = r350173 / r350174;
        double r350176 = acos(r350175);
        return r350176;
}
↓
double f(double v) {
        double r350177 = atan2(1.0, 0.0);
        double r350178 = 2.0;
        double r350179 = r350177 / r350178;
        double r350180 = 4.0;
        double r350181 = v;
        double r350182 = pow(r350181, r350178);
        double r350183 = r350180 * r350182;
        double r350184 = 4.0;
        double r350185 = pow(r350181, r350184);
        double r350186 = r350180 * r350185;
        double r350187 = r350183 + r350186;
        double r350188 = 1.0;
        double r350189 = r350187 - r350188;
        double r350190 = asin(r350189);
        double r350191 = r350179 - r350190;
        return r350191;
}

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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied clear-num0.6

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

  6. Taylor expanded around 0 0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2020020 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))