Average Error: 1.0 → 0.1
Time: 20.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\frac{\sqrt{3} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \left(\sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right) \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right)}{2} + \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{1}{2}\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\frac{\sqrt{3} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \left(\sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right) \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right)}{2} + \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{1}{2}\right) \cdot 2
double f(double g, double h) {
        double r5091169 = 2.0;
        double r5091170 = atan2(1.0, 0.0);
        double r5091171 = r5091169 * r5091170;
        double r5091172 = 3.0;
        double r5091173 = r5091171 / r5091172;
        double r5091174 = g;
        double r5091175 = -r5091174;
        double r5091176 = h;
        double r5091177 = r5091175 / r5091176;
        double r5091178 = acos(r5091177);
        double r5091179 = r5091178 / r5091172;
        double r5091180 = r5091173 + r5091179;
        double r5091181 = cos(r5091180);
        double r5091182 = r5091169 * r5091181;
        return r5091182;
}
↓
double f(double g, double h) {
        double r5091183 = 3.0;
        double r5091184 = sqrt(r5091183);
        double r5091185 = g;
        double r5091186 = h;
        double r5091187 = r5091185 / r5091186;
        double r5091188 = acos(r5091187);
        double r5091189 = r5091188 / r5091183;
        double r5091190 = atan2(1.0, 0.0);
        double r5091191 = 1.5;
        double r5091192 = r5091190 / r5091191;
        double r5091193 = cbrt(r5091192);
        double r5091194 = r5091193 * r5091193;
        double r5091195 = r5091194 * r5091193;
        double r5091196 = r5091189 - r5091195;
        double r5091197 = sin(r5091196);
        double r5091198 = r5091184 * r5091197;
        double r5091199 = 2.0;
        double r5091200 = r5091198 / r5091199;
        double r5091201 = r5091189 - r5091192;
        double r5091202 = cos(r5091201);
        double r5091203 = 0.5;
        double r5091204 = r5091202 * r5091203;
        double r5091205 = r5091200 + r5091204;
        double r5091206 = r5091205 * r5091199;
        return r5091206;
}

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

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

  4. Applied distribute-frac-neg1.0

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

  5. Applied acos-neg1.0

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

  6. Applied div-sub1.0

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

  7. Applied associate-+l-1.0

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

  8. Applied cos-diff0.1

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

  9. Simplified0.1

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

  10. Simplified0.1

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

  12. Applied add-cube-cbrt0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))