2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 10.4s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

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)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

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

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

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]

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

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)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

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

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

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]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{3}}{2}\\ t_1 := \cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\\ t_2 := \left(t\_0 \cdot 0.5\right) \cdot \sin t\_1\\ t_3 := \left(0.25 - t\_0 \cdot t\_0\right) \cdot \cos t\_1\\ 2 \cdot \frac{{t\_3}^{3} - {\left(0.75 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right)\right)}^{1.5}}{{t\_3}^{2} + \left(2 \cdot t\_2\right) \cdot \mathsf{fma}\left(2, t\_2, t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (/ (sqrt 3.0) 2.0))
        (t_1 (* (acos (/ g (- h))) 0.3333333333333333))
        (t_2 (* (* t_0 0.5) (sin t_1)))
        (t_3 (* (- 0.25 (* t_0 t_0)) (cos t_1))))
   (*
    2.0
    (/
     (- (pow t_3 3.0) (pow (* 0.75 (- 0.5 (* 0.5 (cos (* 2.0 t_1))))) 1.5))
     (+ (pow t_3 2.0) (* (* 2.0 t_2) (fma 2.0 t_2 t_3)))))))

C

double code(double g, double h) {
	double t_0 = sqrt(3.0) / 2.0;
	double t_1 = acos((g / -h)) * 0.3333333333333333;
	double t_2 = (t_0 * 0.5) * sin(t_1);
	double t_3 = (0.25 - (t_0 * t_0)) * cos(t_1);
	return 2.0 * ((pow(t_3, 3.0) - pow((0.75 * (0.5 - (0.5 * cos((2.0 * t_1))))), 1.5)) / (pow(t_3, 2.0) + ((2.0 * t_2) * fma(2.0, t_2, t_3))));
}

Julia

function code(g, h)
	t_0 = Float64(sqrt(3.0) / 2.0)
	t_1 = Float64(acos(Float64(g / Float64(-h))) * 0.3333333333333333)
	t_2 = Float64(Float64(t_0 * 0.5) * sin(t_1))
	t_3 = Float64(Float64(0.25 - Float64(t_0 * t_0)) * cos(t_1))
	return Float64(2.0 * Float64(Float64((t_3 ^ 3.0) - (Float64(0.75 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1))))) ^ 1.5)) / Float64((t_3 ^ 2.0) + Float64(Float64(2.0 * t_2) * fma(2.0, t_2, t_3)))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(N[Sqrt[3.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.25 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] - N[Power[N[(0.75 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$3, 2.0], $MachinePrecision] + N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(2.0 * t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - {\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right)}^{3}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - \color{blue}{{\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)}^{3}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   3. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right)}\right)}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   4. pow-sqr N/A

      \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - \color{blue}{{\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right)} \cdot {\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right)}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   5. pow-prod-down N/A

      \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - \color{blue}{{\left(\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right)}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - \color{blue}{{\left(\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right)}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - \color{blue}{{\left(0.75 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right)\right)}^{1.5}}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)} \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{g}{-h}\right)\\ t_1 := \cos \left(\mathsf{fma}\left(\pi, -2, t\_0\right) \cdot -0.3333333333333333\right)\\ t\_1 \cdot \frac{\frac{2}{t\_1}}{\frac{1}{\cos \left(\mathsf{fma}\left(t\_0, 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\right)}} \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ g (- h))))
        (t_1 (cos (* (fma PI -2.0 t_0) -0.3333333333333333))))
   (*
    t_1
    (/
     (/ 2.0 t_1)
     (/ 1.0 (cos (fma t_0 0.3333333333333333 (* PI 0.6666666666666666))))))))

C

double code(double g, double h) {
	double t_0 = acos((g / -h));
	double t_1 = cos((fma(((double) M_PI), -2.0, t_0) * -0.3333333333333333));
	return t_1 * ((2.0 / t_1) / (1.0 / cos(fma(t_0, 0.3333333333333333, (((double) M_PI) * 0.6666666666666666)))));
}

Julia

function code(g, h)
	t_0 = acos(Float64(g / Float64(-h)))
	t_1 = cos(Float64(fma(pi, -2.0, t_0) * -0.3333333333333333))
	return Float64(t_1 * Float64(Float64(2.0 / t_1) / Float64(1.0 / cos(fma(t_0, 0.3333333333333333, Float64(pi * 0.6666666666666666))))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(Pi * -2.0 + t$95$0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, N[(t$95$1 * N[(N[(2.0 / t$95$1), $MachinePrecision] / N[(1.0 / N[Cos[N[(t$95$0 * 0.3333333333333333 + N[(Pi * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\frac{\cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right) \cdot \cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\right)}{\cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-/.f64 N/A

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

6. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{2}{\cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right)} \cdot \left(\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\right) \cdot \cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{\cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right)}}{\frac{1}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\right)}} \cdot \cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right)} \]

9. Final simplification 99.9%

   \[\leadsto \cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right) \cdot \frac{\frac{2}{\cos \left(\mathsf{fma}\left(\pi, -2, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.3333333333333333\right)}}{\frac{1}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\right)}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))