Average Error: 1.0 → 0.0
Time: 19.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \left(\sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \left(\sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)}\right)\right) \cdot 2
double f(double g, double h) {
        double r2485272 = 2.0;
        double r2485273 = atan2(1.0, 0.0);
        double r2485274 = r2485272 * r2485273;
        double r2485275 = 3.0;
        double r2485276 = r2485274 / r2485275;
        double r2485277 = g;
        double r2485278 = -r2485277;
        double r2485279 = h;
        double r2485280 = r2485278 / r2485279;
        double r2485281 = acos(r2485280);
        double r2485282 = r2485281 / r2485275;
        double r2485283 = r2485276 + r2485282;
        double r2485284 = cos(r2485283);
        double r2485285 = r2485272 * r2485284;
        return r2485285;
}
↓
double f(double g, double h) {
        double r2485286 = 0.6666666666666666;
        double r2485287 = atan2(1.0, 0.0);
        double r2485288 = r2485286 * r2485287;
        double r2485289 = cos(r2485288);
        double r2485290 = g;
        double r2485291 = h;
        double r2485292 = r2485290 / r2485291;
        double r2485293 = -r2485292;
        double r2485294 = acos(r2485293);
        double r2485295 = 3.0;
        double r2485296 = sqrt(r2485295);
        double r2485297 = r2485294 / r2485296;
        double r2485298 = r2485297 / r2485296;
        double r2485299 = cos(r2485298);
        double r2485300 = r2485289 * r2485299;
        double r2485301 = sin(r2485298);
        double r2485302 = sin(r2485288);
        double r2485303 = sqrt(r2485302);
        double r2485304 = r2485303 * r2485303;
        double r2485305 = r2485301 * r2485304;
        double r2485306 = r2485300 - r2485305;
        double r2485307 = 2.0;
        double r2485308 = r2485306 * r2485307;
        return r2485308;
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified1.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied associate-/r*1.0

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

  7. Applied fma-udef1.0

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

  8. Applied cos-sum1.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))