2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 9.2s

Alternatives: 3

Speedup: 1.1×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ 2 \cdot \left(-0.5 \cdot \left(\sqrt{3} \cdot \sin \left(t\_0 \cdot 0.3333333333333333\right) + \cos \left(t\_0 \cdot -0.3333333333333333\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (*
    2.0
    (*
     -0.5
     (+
      (* (sqrt 3.0) (sin (* t_0 0.3333333333333333)))
      (cos (* t_0 -0.3333333333333333)))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	return 2.0 * (-0.5 * ((sqrt(3.0) * sin((t_0 * 0.3333333333333333))) + cos((t_0 * -0.3333333333333333))));
}

Fortran

real(8) function code(g, h)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8) :: t_0
    t_0 = acos((-g / h))
    code = 2.0d0 * ((-0.5d0) * ((sqrt(3.0d0) * sin((t_0 * 0.3333333333333333d0))) + cos((t_0 * (-0.3333333333333333d0)))))
end function

Java

public static double code(double g, double h) {
	double t_0 = Math.acos((-g / h));
	return 2.0 * (-0.5 * ((Math.sqrt(3.0) * Math.sin((t_0 * 0.3333333333333333))) + Math.cos((t_0 * -0.3333333333333333))));
}

Python

def code(g, h):
	t_0 = math.acos((-g / h))
	return 2.0 * (-0.5 * ((math.sqrt(3.0) * math.sin((t_0 * 0.3333333333333333))) + math.cos((t_0 * -0.3333333333333333))))

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	return Float64(2.0 * Float64(-0.5 * Float64(Float64(sqrt(3.0) * sin(Float64(t_0 * 0.3333333333333333))) + cos(Float64(t_0 * -0.3333333333333333)))))
end

MATLAB

function tmp = code(g, h)
	t_0 = acos((-g / h));
	tmp = 2.0 * (-0.5 * ((sqrt(3.0) * sin((t_0 * 0.3333333333333333))) + cos((t_0 * -0.3333333333333333))));
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[(-0.5 * N[(N[(N[Sqrt[3.0], $MachinePrecision] * N[Sin[N[(t$95$0 * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Cos[N[(t$95$0 * -0.3333333333333333), $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 egg-rr 99.9%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. metadata-eval N/A

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

   15. distribute-lft-neg-in N/A

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

9. Simplified 99.9%

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

    1. lift-neg.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-acos.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-sin.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-neg.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. lift-acos.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-cos.f64 N/A

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

    12. lower-+.f64 N/A

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

    \[\leadsto 2 \cdot \left(-0.5 \cdot \left(\sqrt{3} \cdot \sin \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot 0.3333333333333333\right) + \cos \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ 2 \cdot \left(-0.5 \cdot \mathsf{fma}\left(\sin \left(t\_0 \cdot 0.3333333333333333\right), \sqrt{3}, \cos \left(t\_0 \cdot -0.3333333333333333\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (*
    2.0
    (*
     -0.5
     (fma
      (sin (* t_0 0.3333333333333333))
      (sqrt 3.0)
      (cos (* t_0 -0.3333333333333333)))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	return 2.0 * (-0.5 * fma(sin((t_0 * 0.3333333333333333)), sqrt(3.0), cos((t_0 * -0.3333333333333333))));
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	return Float64(2.0 * Float64(-0.5 * fma(sin(Float64(t_0 * 0.3333333333333333)), sqrt(3.0), cos(Float64(t_0 * -0.3333333333333333)))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[(-0.5 * N[(N[Sin[N[(t$95$0 * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[N[(t$95$0 * -0.3333333333333333), $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 egg-rr 99.9%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. metadata-eval N/A

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

   15. distribute-lft-neg-in N/A

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

    \[\leadsto 2 \cdot \left(-0.5 \cdot \mathsf{fma}\left(\sin \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot 0.3333333333333333\right), \sqrt{3}, \cos \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(Pi * 0.6666666666666666 + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.5

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

   14. lift-/.f64 N/A

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

   15. div-inv N/A

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

   16. lower-*.f64 N/A

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))