Average Error: 1.0 → 0.0
Time: 21.0s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\left(\left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \cos \left(\left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right) \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right) \cdot \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\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
double f(double g, double h) {
        double r8458026 = 2.0;
        double r8458027 = atan2(1.0, 0.0);
        double r8458028 = r8458026 * r8458027;
        double r8458029 = 3.0;
        double r8458030 = r8458028 / r8458029;
        double r8458031 = g;
        double r8458032 = -r8458031;
        double r8458033 = h;
        double r8458034 = r8458032 / r8458033;
        double r8458035 = acos(r8458034);
        double r8458036 = r8458035 / r8458029;
        double r8458037 = r8458030 + r8458036;
        double r8458038 = cos(r8458037);
        double r8458039 = r8458026 * r8458038;
        return r8458039;
}
↓
double f(double g, double h) {
        double r8458040 = 2.0;
        double r8458041 = 3.0;
        double r8458042 = r8458040 / r8458041;
        double r8458043 = atan2(1.0, 0.0);
        double r8458044 = r8458042 * r8458043;
        double r8458045 = cos(r8458044);
        double r8458046 = cbrt(r8458045);
        double r8458047 = r8458046 * r8458046;
        double r8458048 = r8458047 * r8458046;
        double r8458049 = 1.0;
        double r8458050 = sqrt(r8458041);
        double r8458051 = r8458049 / r8458050;
        double r8458052 = sqrt(r8458051);
        double r8458053 = r8458052 * r8458052;
        double r8458054 = g;
        double r8458055 = -r8458054;
        double r8458056 = h;
        double r8458057 = r8458055 / r8458056;
        double r8458058 = acos(r8458057);
        double r8458059 = r8458058 / r8458050;
        double r8458060 = r8458053 * r8458059;
        double r8458061 = cos(r8458060);
        double r8458062 = r8458048 * r8458061;
        double r8458063 = sin(r8458044);
        double r8458064 = sin(r8458060);
        double r8458065 = r8458063 * r8458064;
        double r8458066 = r8458062 - r8458065;
        double r8458067 = r8458066 * r8458040;
        return r8458067;
}

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 *-un-lft-identity1.0

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

  6. Applied times-frac1.0

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

  8. Applied add-sqr-sqrt1.0

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

  10. Applied fma-udef1.0

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

  11. Applied cos-sum0.0

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

  13. Applied add-cube-cbrt0.0

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

  14. Final simplification0.0

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

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)))))