Average Error: 1.0 → 0.0
Time: 13.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)
double f(double g, double h) {
        double r200220 = 2.0;
        double r200221 = atan2(1.0, 0.0);
        double r200222 = r200220 * r200221;
        double r200223 = 3.0;
        double r200224 = r200222 / r200223;
        double r200225 = g;
        double r200226 = -r200225;
        double r200227 = h;
        double r200228 = r200226 / r200227;
        double r200229 = acos(r200228);
        double r200230 = r200229 / r200223;
        double r200231 = r200224 + r200230;
        double r200232 = cos(r200231);
        double r200233 = r200220 * r200232;
        return r200233;
}
↓
double f(double g, double h) {
        double r200234 = 2.0;
        double r200235 = g;
        double r200236 = h;
        double r200237 = r200235 / r200236;
        double r200238 = acos(r200237);
        double r200239 = 3.0;
        double r200240 = r200238 / r200239;
        double r200241 = cos(r200240);
        double r200242 = r200234 / r200239;
        double r200243 = atan2(1.0, 0.0);
        double r200244 = r200243 / r200239;
        double r200245 = fma(r200242, r200243, r200244);
        double r200246 = cos(r200245);
        double r200247 = r200241 * r200246;
        double r200248 = sin(r200245);
        double r200249 = sin(r200240);
        double r200250 = r200248 * r200249;
        double r200251 = r200247 + r200250;
        double r200252 = r200234 * r200251;
        return r200252;
}

Error

Bits error versus g

Bits error versus h

Derivation

  1. Initial program 1.0

    \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
  2. Using strategy rm

  3. Applied distribute-frac-neg1.0

    \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \color{blue}{\left(-\frac{g}{h}\right)}}{3}\right)\]

  4. Applied acos-neg1.0

    \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}}{3}\right)\]

  5. Applied div-sub1.0

    \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \color{blue}{\left(\frac{\pi}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\]

  6. Applied associate-+r-1.0

    \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\]

  7. Applied cos-diff0.0

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}\]

  8. Simplified0.0

    \[\leadsto 2 \cdot \left(\color{blue}{\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right)} + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  9. Simplified0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \color{blue}{\sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\]

  10. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\pi}{3}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))