Average Error: 1.0 → 0.0
Time: 5.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) - \sin \left(\frac{2}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\]
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{2}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) - \sin \left(\frac{2}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)
double f(double g, double h) {
        double r137292 = 2.0;
        double r137293 = atan2(1.0, 0.0);
        double r137294 = r137292 * r137293;
        double r137295 = 3.0;
        double r137296 = r137294 / r137295;
        double r137297 = g;
        double r137298 = -r137297;
        double r137299 = h;
        double r137300 = r137298 / r137299;
        double r137301 = acos(r137300);
        double r137302 = r137301 / r137295;
        double r137303 = r137296 + r137302;
        double r137304 = cos(r137303);
        double r137305 = r137292 * r137304;
        return r137305;
}
↓
double f(double g, double h) {
        double r137306 = 2.0;
        double r137307 = 3.0;
        double r137308 = cbrt(r137307);
        double r137309 = r137308 * r137308;
        double r137310 = r137306 / r137309;
        double r137311 = atan2(1.0, 0.0);
        double r137312 = r137311 / r137308;
        double r137313 = r137310 * r137312;
        double r137314 = cos(r137313);
        double r137315 = g;
        double r137316 = -r137315;
        double r137317 = h;
        double r137318 = r137316 / r137317;
        double r137319 = acos(r137318);
        double r137320 = r137319 / r137309;
        double r137321 = r137320 / r137308;
        double r137322 = cos(r137321);
        double r137323 = r137314 * r137322;
        double r137324 = sin(r137313);
        double r137325 = r137319 / r137307;
        double r137326 = sin(r137325);
        double r137327 = r137324 * r137326;
        double r137328 = r137323 - r137327;
        double r137329 = r137306 * r137328;
        return r137329;
}

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. Using strategy rm

  3. Applied add-cube-cbrt1.0

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

  4. Applied times-frac1.0

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

  5. Applied fma-def1.0

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

  7. Applied fma-udef1.0

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

  8. Applied cos-sum1.0

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

  10. Applied add-cube-cbrt1.0

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

  11. Applied associate-/r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))