2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 20.5s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r3959246 = 2.0;
        double r3959247 = atan2(1.0, 0.0);
        double r3959248 = r3959246 * r3959247;
        double r3959249 = 3.0;
        double r3959250 = r3959248 / r3959249;
        double r3959251 = g;
        double r3959252 = -r3959251;
        double r3959253 = h;
        double r3959254 = r3959252 / r3959253;
        double r3959255 = acos(r3959254);
        double r3959256 = r3959255 / r3959249;
        double r3959257 = r3959250 + r3959256;
        double r3959258 = cos(r3959257);
        double r3959259 = r3959246 * r3959258;
        return r3959259;
}

↓

double f(double g, double h) {
        double r3959260 = 2.0;
        double r3959261 = 1.0;
        double r3959262 = 3.0;
        double r3959263 = sqrt(r3959262);
        double r3959264 = r3959261 / r3959263;
        double r3959265 = sqrt(r3959264);
        double r3959266 = r3959265 * r3959265;
        double r3959267 = g;
        double r3959268 = h;
        double r3959269 = r3959267 / r3959268;
        double r3959270 = -r3959269;
        double r3959271 = acos(r3959270);
        double r3959272 = r3959271 / r3959263;
        double r3959273 = r3959266 * r3959272;
        double r3959274 = cos(r3959273);
        double r3959275 = r3959260 / r3959262;
        double r3959276 = atan2(1.0, 0.0);
        double r3959277 = r3959275 * r3959276;
        double r3959278 = cos(r3959277);
        double r3959279 = cbrt(r3959278);
        double r3959280 = r3959279 * r3959279;
        double r3959281 = r3959280 * r3959279;
        double r3959282 = r3959274 * r3959281;
        double r3959283 = sin(r3959273);
        double r3959284 = sin(r3959277);
        double r3959285 = r3959283 * r3959284;
        double r3959286 = r3959282 - r3959285;
        double r3959287 = r3959260 * r3959286;
        return r3959287;
}

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. Simplified 1.0

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

4. Applied add-sqr-sqrt 1.0

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

5. Applied *-un-lft-identity 1.0

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

6. Applied times-frac 1.0

   \[\leadsto \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}\right)\right) \cdot 2\]
7. Using strategy rm

8. Applied add-sqr-sqrt 1.0

   \[\leadsto \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right)} \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right) \cdot 2\]
9. Using strategy rm

10. Applied fma-udef 1.0

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

11. Applied cos-sum 0.0

    \[\leadsto \color{blue}{\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right) \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right) \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right)} \cdot 2\]
12. Using strategy rm

13. Applied add-cube-cbrt 0.0

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

14. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))