2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 21.9s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r3903006 = 2.0;
        double r3903007 = atan2(1.0, 0.0);
        double r3903008 = r3903006 * r3903007;
        double r3903009 = 3.0;
        double r3903010 = r3903008 / r3903009;
        double r3903011 = g;
        double r3903012 = -r3903011;
        double r3903013 = h;
        double r3903014 = r3903012 / r3903013;
        double r3903015 = acos(r3903014);
        double r3903016 = r3903015 / r3903009;
        double r3903017 = r3903010 + r3903016;
        double r3903018 = cos(r3903017);
        double r3903019 = r3903006 * r3903018;
        return r3903019;
}

↓

double f(double g, double h) {
        double r3903020 = 2.0;
        double r3903021 = 0.6666666666666666;
        double r3903022 = atan2(1.0, 0.0);
        double r3903023 = g;
        double r3903024 = h;
        double r3903025 = r3903023 / r3903024;
        double r3903026 = -r3903025;
        double r3903027 = acos(r3903026);
        double r3903028 = 3.0;
        double r3903029 = r3903027 / r3903028;
        double r3903030 = fma(r3903021, r3903022, r3903029);
        double r3903031 = cos(r3903030);
        double r3903032 = exp(r3903031);
        double r3903033 = cbrt(r3903032);
        double r3903034 = r3903033 * r3903033;
        double r3903035 = log(r3903034);
        double r3903036 = cbrt(r3903029);
        double r3903037 = r3903036 * r3903036;
        double r3903038 = r3903036 * r3903037;
        double r3903039 = fma(r3903021, r3903022, r3903038);
        double r3903040 = cos(r3903039);
        double r3903041 = exp(r3903040);
        double r3903042 = cbrt(r3903041);
        double r3903043 = log(r3903042);
        double r3903044 = r3903035 + r3903043;
        double r3903045 = r3903020 * r3903044;
        return r3903045;
}

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-log-exp 1.0

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

6. Applied add-cube-cbrt 0.1

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

7. Applied log-prod 0.1

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

9. Applied add-cube-cbrt 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))