2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 13.9s

Alternatives: 3

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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\\ -\mathsf{fma}\left(\sin t\_0, \sqrt{3}, \cos t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (- (/ g h))))))
   (- (fma (sin t_0) (sqrt 3.0) (cos t_0)))))

C

double code(double g, double h) {
	double t_0 = 0.3333333333333333 * acos(-(g / h));
	return -fma(sin(t_0), sqrt(3.0), cos(t_0));
}

Julia

function code(g, h)
	t_0 = Float64(0.3333333333333333 * acos(Float64(-Float64(g / h))))
	return Float64(-fma(sin(t_0), sqrt(3.0), cos(t_0)))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, (-N[(N[Sin[t$95$0], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[t$95$0], $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
3. Step-by-step derivation

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right) + \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{4} \cdot {\left(\sqrt{3}\right)}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. unpow2 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-out N/A

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

   10. associate-*r* N/A

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

7. Simplified 99.9%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right)} \]
8. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right) \cdot \left(\pi \cdot -0.2222222222222222\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos
   (* (fma (acos (- (/ g h))) (/ -1.5 PI) -3.0) (* PI -0.2222222222222222)))
  2.0))

C

double code(double g, double h) {
	return cos((fma(acos(-(g / h)), (-1.5 / ((double) M_PI)), -3.0) * (((double) M_PI) * -0.2222222222222222))) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(Float64(fma(acos(Float64(-Float64(g / h))), Float64(-1.5 / pi), -3.0) * Float64(pi * -0.2222222222222222))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] * N[(-1.5 / Pi), $MachinePrecision] + -3.0), $MachinePrecision] * N[(Pi * -0.2222222222222222), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $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
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. frac-add N/A

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

   6. /-lowering-/.f64 N/A

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

4. Applied egg-rr 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1.5}{\pi}, -3\right)}{3 \cdot \frac{-1.5}{\pi}}\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right) \cdot \left(-0.2222222222222222 \cdot \pi\right)\right) \cdot 2} \]

7. Final simplification 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right) \cdot \left(\pi \cdot -0.2222222222222222\right)\right) \cdot 2 \]
8. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(0.3333333333333333 \cdot \mathsf{fma}\left(2, \pi, \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (* 0.3333333333333333 (fma 2.0 PI (acos (- (/ g h))))))))

C

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

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(0.3333333333333333 * fma(2.0, pi, acos(Float64(-Float64(g / h)))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(0.3333333333333333 * N[(2.0 * Pi + N[ArcCos[(-N[(g / h), $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
3. Step-by-step derivation

   1. div-inv N/A

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

   2. div-inv N/A

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

   3. distribute-rgt-out N/A

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

   4. *-lowering-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. PI-lowering-PI.f64 N/A

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

   8. acos-lowering-acos.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. neg-lowering-neg.f64 98.5

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))