2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 5.1s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r134011 = 2.0;
        double r134012 = atan2(1.0, 0.0);
        double r134013 = r134011 * r134012;
        double r134014 = 3.0;
        double r134015 = r134013 / r134014;
        double r134016 = g;
        double r134017 = -r134016;
        double r134018 = h;
        double r134019 = r134017 / r134018;
        double r134020 = acos(r134019);
        double r134021 = r134020 / r134014;
        double r134022 = r134015 + r134021;
        double r134023 = cos(r134022);
        double r134024 = r134011 * r134023;
        return r134024;
}

↓

double f(double g, double h) {
        double r134025 = 2.0;
        double r134026 = 3.0;
        double r134027 = cbrt(r134026);
        double r134028 = r134027 * r134027;
        double r134029 = r134025 / r134028;
        double r134030 = atan2(1.0, 0.0);
        double r134031 = r134030 / r134027;
        double r134032 = g;
        double r134033 = -r134032;
        double r134034 = h;
        double r134035 = r134033 / r134034;
        double r134036 = acos(r134035);
        double r134037 = r134036 / r134026;
        double r134038 = fma(r134029, r134031, r134037);
        double r134039 = cos(r134038);
        double r134040 = r134039 * r134039;
        double r134041 = cbrt(r134040);
        double r134042 = cbrt(r134039);
        double r134043 = r134041 * r134042;
        double r134044 = 3.0;
        double r134045 = pow(r134043, r134044);
        double r134046 = cbrt(r134045);
        double r134047 = r134025 * r134046;
        return r134047;
}

Error

Bits error versus g

Bits error versus h

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-cbrt 1.0

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

4. Applied times-frac 1.0

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

5. Applied fma-def 1.0

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

7. Applied add-cbrt-cube 1.0

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

8. Simplified 1.0

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

10. Applied add-cube-cbrt 1.0

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))