2-ancestry mixing, negative discriminant

Average Error: 1.0 → 1.0

Time: 23.7s

Precision: 64

Math

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

↓

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right) \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right)\right)\right) \cdot 2\]

C

double f(double g, double h) {
        double r2114047 = 2.0;
        double r2114048 = atan2(1.0, 0.0);
        double r2114049 = r2114047 * r2114048;
        double r2114050 = 3.0;
        double r2114051 = r2114049 / r2114050;
        double r2114052 = g;
        double r2114053 = -r2114052;
        double r2114054 = h;
        double r2114055 = r2114053 / r2114054;
        double r2114056 = acos(r2114055);
        double r2114057 = r2114056 / r2114050;
        double r2114058 = r2114051 + r2114057;
        double r2114059 = cos(r2114058);
        double r2114060 = r2114047 * r2114059;
        return r2114060;
}

↓

double f(double g, double h) {
        double r2114061 = 0.6666666666666666;
        double r2114062 = atan2(1.0, 0.0);
        double r2114063 = 1.0;
        double r2114064 = 3.0;
        double r2114065 = sqrt(r2114064);
        double r2114066 = r2114063 / r2114065;
        double r2114067 = r2114066 * r2114066;
        double r2114068 = g;
        double r2114069 = h;
        double r2114070 = r2114068 / r2114069;
        double r2114071 = -r2114070;
        double r2114072 = acos(r2114071);
        double r2114073 = r2114067 * r2114072;
        double r2114074 = fma(r2114061, r2114062, r2114073);
        double r2114075 = cos(r2114074);
        double r2114076 = expm1(r2114075);
        double r2114077 = log1p(r2114076);
        double r2114078 = 2.0;
        double r2114079 = r2114077 * r2114078;
        return r2114079;
}

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. Simplified 1.0

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

4. Applied add-sqr-sqrt 1.0

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

5. Applied associate-/r* 1.0

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

7. Applied log1p-expm1-u 1.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right)\right)\right)\right)} \cdot 2\]

8. Simplified 1.0

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right) \cdot 2\]
9. Using strategy rm

10. Applied add-sqr-sqrt 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right)\right)\right)\right) \cdot 2\]

11. Applied add-sqr-sqrt 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\color{blue}{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}}{\sqrt{3} \cdot \sqrt{3}}\right)\right)\right)\right) \cdot 2\]

12. Applied times-frac 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}}\right)\right)\right)\right) \cdot 2\]
13. Using strategy rm

14. Applied div-inv 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \frac{1}{\sqrt{3}}\right)}\right)\right)\right)\right) \cdot 2\]

15. Applied div-inv 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\left(\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \frac{1}{\sqrt{3}}\right)} \cdot \left(\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \frac{1}{\sqrt{3}}\right)\right)\right)\right)\right) \cdot 2\]

16. Applied swap-sqr 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\left(\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right) \cdot \left(\frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right)}\right)\right)\right)\right) \cdot 2\]

17. Simplified 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right)} \cdot \left(\frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right)\right)\right)\right)\right) \cdot 2\]

18. Final simplification 1.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}\right) \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right)\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))