Average Error: 1.0 → 0.0
Time: 18.7s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right) - \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{2}{3}\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right)\right) \cdot 2\]
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{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right) - \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{2}{3}\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right)\right) \cdot 2
double f(double g, double h) {
        double r5183204 = 2.0;
        double r5183205 = atan2(1.0, 0.0);
        double r5183206 = r5183204 * r5183205;
        double r5183207 = 3.0;
        double r5183208 = r5183206 / r5183207;
        double r5183209 = g;
        double r5183210 = -r5183209;
        double r5183211 = h;
        double r5183212 = r5183210 / r5183211;
        double r5183213 = acos(r5183212);
        double r5183214 = r5183213 / r5183207;
        double r5183215 = r5183208 + r5183214;
        double r5183216 = cos(r5183215);
        double r5183217 = r5183204 * r5183216;
        return r5183217;
}
↓
double f(double g, double h) {
        double r5183218 = 0.6666666666666666;
        double r5183219 = atan2(1.0, 0.0);
        double r5183220 = r5183218 * r5183219;
        double r5183221 = cos(r5183220);
        double r5183222 = g;
        double r5183223 = h;
        double r5183224 = r5183222 / r5183223;
        double r5183225 = -r5183224;
        double r5183226 = acos(r5183225);
        double r5183227 = sqrt(r5183226);
        double r5183228 = 3.0;
        double r5183229 = r5183228 / r5183227;
        double r5183230 = r5183227 / r5183229;
        double r5183231 = cos(r5183230);
        double r5183232 = r5183221 * r5183231;
        double r5183233 = sqrt(r5183219);
        double r5183234 = r5183233 * r5183218;
        double r5183235 = r5183233 * r5183234;
        double r5183236 = sin(r5183235);
        double r5183237 = sin(r5183230);
        double r5183238 = r5183236 * r5183237;
        double r5183239 = r5183232 - r5183238;
        double r5183240 = 2.0;
        double r5183241 = r5183239 * r5183240;
        return r5183241;
}

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

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

  5. Applied associate-/l*1.0

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

  7. Applied fma-udef1.0

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

  8. Applied cos-sum1.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied associate-*r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))