Average Error: 1.0 → 0.0
Time: 3.9s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\frac{2}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\]
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}}}{\sqrt{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 r132008 = 2.0;
        double r132009 = atan2(1.0, 0.0);
        double r132010 = r132008 * r132009;
        double r132011 = 3.0;
        double r132012 = r132010 / r132011;
        double r132013 = g;
        double r132014 = -r132013;
        double r132015 = h;
        double r132016 = r132014 / r132015;
        double r132017 = acos(r132016);
        double r132018 = r132017 / r132011;
        double r132019 = r132012 + r132018;
        double r132020 = cos(r132019);
        double r132021 = r132008 * r132020;
        return r132021;
}
↓
double f(double g, double h) {
        double r132022 = 2.0;
        double r132023 = 3.0;
        double r132024 = cbrt(r132023);
        double r132025 = r132024 * r132024;
        double r132026 = r132022 / r132025;
        double r132027 = atan2(1.0, 0.0);
        double r132028 = r132027 / r132024;
        double r132029 = r132026 * r132028;
        double r132030 = cos(r132029);
        double r132031 = g;
        double r132032 = -r132031;
        double r132033 = h;
        double r132034 = r132032 / r132033;
        double r132035 = acos(r132034);
        double r132036 = sqrt(r132023);
        double r132037 = r132035 / r132036;
        double r132038 = r132037 / r132036;
        double r132039 = cos(r132038);
        double r132040 = r132030 * r132039;
        double r132041 = sin(r132029);
        double r132042 = r132035 / r132023;
        double r132043 = sin(r132042);
        double r132044 = r132041 * r132043;
        double r132045 = r132040 - r132044;
        double r132046 = r132022 * r132045;
        return r132046;
}

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-sqr-sqrt0.0

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

  11. Applied associate-/r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))