Average Error: 0.6 → 0.6
Time: 12.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\frac{\mathsf{fma}\left(1, 1, -\left(5 \cdot 5\right) \cdot {v}^{4}\right)}{\mathsf{fma}\left(v \cdot v, 5, 1\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\frac{\mathsf{fma}\left(1, 1, -\left(5 \cdot 5\right) \cdot {v}^{4}\right)}{\mathsf{fma}\left(v \cdot v, 5, 1\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r222472 = 1.0;
        double r222473 = 5.0;
        double r222474 = v;
        double r222475 = r222474 * r222474;
        double r222476 = r222473 * r222475;
        double r222477 = r222472 - r222476;
        double r222478 = r222475 - r222472;
        double r222479 = r222477 / r222478;
        double r222480 = acos(r222479);
        return r222480;
}
↓
double f(double v) {
        double r222481 = 1.0;
        double r222482 = 5.0;
        double r222483 = r222482 * r222482;
        double r222484 = v;
        double r222485 = 4.0;
        double r222486 = pow(r222484, r222485);
        double r222487 = r222483 * r222486;
        double r222488 = -r222487;
        double r222489 = fma(r222481, r222481, r222488);
        double r222490 = r222484 * r222484;
        double r222491 = fma(r222490, r222482, r222481);
        double r222492 = r222489 / r222491;
        double r222493 = r222490 - r222481;
        double r222494 = r222492 / r222493;
        double r222495 = acos(r222494);
        return r222495;
}

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. 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. Simplified0.6

    \[\leadsto \cos^{-1} \left(\frac{\frac{\color{blue}{\mathsf{fma}\left(1, 1, -\left(5 \cdot 5\right) \cdot {v}^{4}\right)}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

  5. Simplified0.6

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

  6. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\frac{\frac{\mathsf{fma}\left(1, 1, -\left(5 \cdot 5\right) \cdot {v}^{4}\right)}{\mathsf{fma}\left(v \cdot v, 5, 1\right)}}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))