Average Error: 1.0 → 0.1
Time: 22.4s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(2 \cdot \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) + \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)
double f(double g, double h) {
        double r104929 = 2.0;
        double r104930 = atan2(1.0, 0.0);
        double r104931 = r104929 * r104930;
        double r104932 = 3.0;
        double r104933 = r104931 / r104932;
        double r104934 = g;
        double r104935 = -r104934;
        double r104936 = h;
        double r104937 = r104935 / r104936;
        double r104938 = acos(r104937);
        double r104939 = r104938 / r104932;
        double r104940 = r104933 + r104939;
        double r104941 = cos(r104940);
        double r104942 = r104929 * r104941;
        return r104942;
}
↓
double f(double g, double h) {
        double r104943 = 2.0;
        double r104944 = 2.0;
        double r104945 = 3.0;
        double r104946 = r104943 / r104945;
        double r104947 = atan2(1.0, 0.0);
        double r104948 = g;
        double r104949 = -r104948;
        double r104950 = h;
        double r104951 = r104949 / r104950;
        double r104952 = acos(r104951);
        double r104953 = r104952 / r104945;
        double r104954 = fma(r104946, r104947, r104953);
        double r104955 = cos(r104954);
        double r104956 = exp(r104955);
        double r104957 = cbrt(r104956);
        double r104958 = log(r104957);
        double r104959 = r104944 * r104958;
        double r104960 = r104959 + r104958;
        double r104961 = r104943 * r104960;
        return r104961;
}

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. Simplified1.0

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

  4. Applied add-log-exp1.0

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied log-prod0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))