Average Error: 1.0 → 0.0
Time: 6.5s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{1}{\sqrt{3}} \cdot \frac{1}{\frac{\sqrt{3}}{\cos^{-1} \left(\frac{-g}{h}\right)}}\\ 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666\right) \cdot \cos t_0 - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.6666666666666666\right)\right)\right) \cdot \sin t_0\right) \end{array} \]
(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 (* (/ 1.0 (sqrt 3.0)) (/ 1.0 (/ (sqrt 3.0) (acos (/ (- g) h)))))))
   (*
    2.0
    (-
     (* (cos (* PI 0.6666666666666666)) (cos t_0))
     (* (sin (expm1 (log1p (* PI 0.6666666666666666)))) (sin 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 = (1.0 / sqrt(3.0)) * (1.0 / (sqrt(3.0) / acos((-g / h))));
	return 2.0 * ((cos((((double) M_PI) * 0.6666666666666666)) * cos(t_0)) - (sin(expm1(log1p((((double) M_PI) * 0.6666666666666666)))) * sin(t_0)));
}

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}
↓
public static double code(double g, double h) {
	double t_0 = (1.0 / Math.sqrt(3.0)) * (1.0 / (Math.sqrt(3.0) / Math.acos((-g / h))));
	return 2.0 * ((Math.cos((Math.PI * 0.6666666666666666)) * Math.cos(t_0)) - (Math.sin(Math.expm1(Math.log1p((Math.PI * 0.6666666666666666)))) * Math.sin(t_0)));
}

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))
↓
def code(g, h):
	t_0 = (1.0 / math.sqrt(3.0)) * (1.0 / (math.sqrt(3.0) / math.acos((-g / h))))
	return 2.0 * ((math.cos((math.pi * 0.6666666666666666)) * math.cos(t_0)) - (math.sin(math.expm1(math.log1p((math.pi * 0.6666666666666666)))) * math.sin(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 = Float64(Float64(1.0 / sqrt(3.0)) * Float64(1.0 / Float64(sqrt(3.0) / acos(Float64(Float64(-g) / h)))))
	return Float64(2.0 * Float64(Float64(cos(Float64(pi * 0.6666666666666666)) * cos(t_0)) - Float64(sin(expm1(log1p(Float64(pi * 0.6666666666666666)))) * sin(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[(N[(1.0 / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[3.0], $MachinePrecision] / N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(N[Cos[N[(Pi * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(Exp[N[Log[1 + N[(Pi * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $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 := \frac{1}{\sqrt{3}} \cdot \frac{1}{\frac{\sqrt{3}}{\cos^{-1} \left(\frac{-g}{h}\right)}}\\
2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666\right) \cdot \cos t_0 - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.6666666666666666\right)\right)\right) \cdot \sin t_0\right)
\end{array}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 clear-num_binary641.0

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

  4. Applied *-un-lft-identity_binary641.0

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

  5. Applied add-sqr-sqrt_binary641.0

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

  6. Applied times-frac_binary641.0

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

  7. Applied *-un-lft-identity_binary641.0

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

  8. Applied times-frac_binary641.0

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

  9. Applied fma-udef_binary641.0

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

  10. Applied cos-sum_binary641.0

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

  11. Applied expm1-log1p-u_binary640.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666\right) \cdot \cos \left(\frac{1}{\frac{\sqrt{3}}{1}} \cdot \frac{1}{\frac{\sqrt{3}}{\cos^{-1} \left(\frac{-g}{h}\right)}}\right) - \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.6666666666666666\right)\right)\right)} \cdot \sin \left(\frac{1}{\frac{\sqrt{3}}{1}} \cdot \frac{1}{\frac{\sqrt{3}}{\cos^{-1} \left(\frac{-g}{h}\right)}}\right)\right) \]

  12. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666\right) \cdot \cos \left(\frac{1}{\sqrt{3}} \cdot \frac{1}{\frac{\sqrt{3}}{\cos^{-1} \left(\frac{-g}{h}\right)}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.6666666666666666\right)\right)\right) \cdot \sin \left(\frac{1}{\sqrt{3}} \cdot \frac{1}{\frac{\sqrt{3}}{\cos^{-1} \left(\frac{-g}{h}\right)}}\right)\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))