Average Error: 1.0 → 0.0
Time: 21.6s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right)\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right)\right)\]
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{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right)\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right)\right)
double f(double g, double h) {
        double r4944015 = 2.0;
        double r4944016 = atan2(1.0, 0.0);
        double r4944017 = r4944015 * r4944016;
        double r4944018 = 3.0;
        double r4944019 = r4944017 / r4944018;
        double r4944020 = g;
        double r4944021 = -r4944020;
        double r4944022 = h;
        double r4944023 = r4944021 / r4944022;
        double r4944024 = acos(r4944023);
        double r4944025 = r4944024 / r4944018;
        double r4944026 = r4944019 + r4944025;
        double r4944027 = cos(r4944026);
        double r4944028 = r4944015 * r4944027;
        return r4944028;
}
↓
double f(double g, double h) {
        double r4944029 = 2.0;
        double r4944030 = g;
        double r4944031 = h;
        double r4944032 = r4944030 / r4944031;
        double r4944033 = -r4944032;
        double r4944034 = acos(r4944033);
        double r4944035 = 3.0;
        double r4944036 = sqrt(r4944035);
        double r4944037 = r4944034 / r4944036;
        double r4944038 = 1.0;
        double r4944039 = r4944038 / r4944036;
        double r4944040 = r4944037 * r4944039;
        double r4944041 = cos(r4944040);
        double r4944042 = 0.6666666666666666;
        double r4944043 = atan2(1.0, 0.0);
        double r4944044 = r4944042 * r4944043;
        double r4944045 = cos(r4944044);
        double r4944046 = cbrt(r4944045);
        double r4944047 = r4944046 * r4944046;
        double r4944048 = r4944046 * r4944047;
        double r4944049 = r4944041 * r4944048;
        double r4944050 = sin(r4944044);
        double r4944051 = sin(r4944040);
        double r4944052 = r4944050 * r4944051;
        double r4944053 = r4944049 - r4944052;
        double r4944054 = r4944029 * r4944053;
        return r4944054;
}

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 fma-udef1.0

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

  9. Applied cos-sum1.0

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

  11. Applied add-cube-cbrt0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))