Average Error: 1.0 → 1.3
Time: 3.8s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}}\\ 2 \cdot \left(\log \left(t_0 \cdot t_0\right) + \log t_0\right) \end{array} \]
(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))
↓
(FPCore (g h)
 :precision binary64
 (let* ((t_0
         (cbrt
          (exp (cos (fma PI 0.6666666666666666 (/ (acos (/ g h)) 3.0)))))))
   (* 2.0 (+ (log (* t_0 t_0)) (log t_0)))))
double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}
↓
double code(double g, double h) {
	double t_0 = cbrt(exp(cos(fma(((double) M_PI), 0.6666666666666666, (acos((g / h)) / 3.0)))));
	return 2.0 * (log((t_0 * t_0)) + log(t_0));
}

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end
↓
function code(g, h)
	t_0 = cbrt(exp(cos(fma(pi, 0.6666666666666666, Float64(acos(Float64(g / h)) / 3.0)))))
	return Float64(2.0 * Float64(log(Float64(t_0 * t_0)) + log(t_0)))
end

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[g_, h_] := Block[{t$95$0 = N[Power[N[Exp[N[Cos[N[(Pi * 0.6666666666666666 + N[(N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(2.0 * N[(N[Log[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\begin{array}{l}
t_0 := \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}}\\
2 \cdot \left(\log \left(t_0 \cdot t_0\right) + \log t_0\right)
\end{array}

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

    \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \]

  3. Applied egg-rr1.3

    \[\leadsto 2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}}\right)\right)} \]

  4. Final simplification1.3

    \[\leadsto 2 \cdot \left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}}\right)\right) \]

Reproduce

herbie shell --seed 2022145 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))