Average Error: 0.6 → 0.8
Time: 25.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v \cdot v, v \cdot v, v \cdot v\right), -1\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v \cdot v, v \cdot v, v \cdot v\right), -1\right)\right)
double f(double v) {
        double r4693056 = 1.0;
        double r4693057 = 5.0;
        double r4693058 = v;
        double r4693059 = r4693058 * r4693058;
        double r4693060 = r4693057 * r4693059;
        double r4693061 = r4693056 - r4693060;
        double r4693062 = r4693059 - r4693056;
        double r4693063 = r4693061 / r4693062;
        double r4693064 = acos(r4693063);
        return r4693064;
}
↓
double f(double v) {
        double r4693065 = atan2(1.0, 0.0);
        double r4693066 = 2.0;
        double r4693067 = r4693065 / r4693066;
        double r4693068 = 4.0;
        double r4693069 = v;
        double r4693070 = r4693069 * r4693069;
        double r4693071 = fma(r4693070, r4693070, r4693070);
        double r4693072 = -1.0;
        double r4693073 = fma(r4693068, r4693071, r4693072);
        double r4693074 = asin(r4693073);
        double r4693075 = r4693067 - r4693074;
        return r4693075;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.6

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

  2. Simplified0.6

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

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  6. Applied acos-asin0.8

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

  7. Final simplification0.8

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

Reproduce

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