2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 11.9s

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(0 - \frac{g}{h}\right)\\ t_1 := \cos \left(\frac{3}{2 \cdot \pi - t\_0} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{t\_0}^{2}}{9}\right)\right)\\ 2 \cdot \left(\left(\cos \left(0.3333333333333333 \cdot \left(t\_0 + 2 \cdot \pi\right)\right) \cdot t\_1\right) \cdot \frac{1}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (- 0.0 (/ g h))))
        (t_1
         (cos
          (*
           (/ 3.0 (- (* 2.0 PI) t_0))
           (+ (* PI (* PI 0.4444444444444444)) (/ (pow t_0 2.0) 9.0))))))
   (*
    2.0
    (* (* (cos (* 0.3333333333333333 (+ t_0 (* 2.0 PI)))) t_1) (/ 1.0 t_1)))))

C

double code(double g, double h) {
	double t_0 = acos((0.0 - (g / h)));
	double t_1 = cos(((3.0 / ((2.0 * ((double) M_PI)) - t_0)) * ((((double) M_PI) * (((double) M_PI) * 0.4444444444444444)) + (pow(t_0, 2.0) / 9.0))));
	return 2.0 * ((cos((0.3333333333333333 * (t_0 + (2.0 * ((double) M_PI))))) * t_1) * (1.0 / t_1));
}

Java

public static double code(double g, double h) {
	double t_0 = Math.acos((0.0 - (g / h)));
	double t_1 = Math.cos(((3.0 / ((2.0 * Math.PI) - t_0)) * ((Math.PI * (Math.PI * 0.4444444444444444)) + (Math.pow(t_0, 2.0) / 9.0))));
	return 2.0 * ((Math.cos((0.3333333333333333 * (t_0 + (2.0 * Math.PI)))) * t_1) * (1.0 / t_1));
}

Python

def code(g, h):
	t_0 = math.acos((0.0 - (g / h)))
	t_1 = math.cos(((3.0 / ((2.0 * math.pi) - t_0)) * ((math.pi * (math.pi * 0.4444444444444444)) + (math.pow(t_0, 2.0) / 9.0))))
	return 2.0 * ((math.cos((0.3333333333333333 * (t_0 + (2.0 * math.pi)))) * t_1) * (1.0 / t_1))

Julia

function code(g, h)
	t_0 = acos(Float64(0.0 - Float64(g / h)))
	t_1 = cos(Float64(Float64(3.0 / Float64(Float64(2.0 * pi) - t_0)) * Float64(Float64(pi * Float64(pi * 0.4444444444444444)) + Float64((t_0 ^ 2.0) / 9.0))))
	return Float64(2.0 * Float64(Float64(cos(Float64(0.3333333333333333 * Float64(t_0 + Float64(2.0 * pi)))) * t_1) * Float64(1.0 / t_1)))
end

MATLAB

function tmp = code(g, h)
	t_0 = acos((0.0 - (g / h)));
	t_1 = cos(((3.0 / ((2.0 * pi) - t_0)) * ((pi * (pi * 0.4444444444444444)) + ((t_0 ^ 2.0) / 9.0))));
	tmp = 2.0 * ((cos((0.3333333333333333 * (t_0 + (2.0 * pi)))) * t_1) * (1.0 / t_1));
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(0.0 - N[(g / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(3.0 / N[(N[(2.0 * Pi), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * N[(Pi * 0.4444444444444444), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[(N[(N[Cos[N[(0.3333333333333333 * N[(t$95$0 + N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 / t$95$1), $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. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right), \color{blue}{2}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\cos \left(0.3333333333333333 \cdot \left(2 \cdot \pi + \cos^{-1} \left(0 - \frac{g}{h}\right)\right)\right) \cdot 2} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\left(\cos \left(0.3333333333333333 \cdot \left(\cos^{-1} \left(0 - \frac{g}{h}\right) + 2 \cdot \pi\right)\right) \cdot \cos \left(\frac{3}{2 \cdot \pi - \cos^{-1} \left(0 - \frac{g}{h}\right)} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9}\right)\right)\right) \cdot \frac{1}{\cos \left(\frac{3}{2 \cdot \pi - \cos^{-1} \left(0 - \frac{g}{h}\right)} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9}\right)\right)}\right)} \cdot 2 \]

7. Final simplification 100.0%

   \[\leadsto 2 \cdot \left(\left(\cos \left(0.3333333333333333 \cdot \left(\cos^{-1} \left(0 - \frac{g}{h}\right) + 2 \cdot \pi\right)\right) \cdot \cos \left(\frac{3}{2 \cdot \pi - \cos^{-1} \left(0 - \frac{g}{h}\right)} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9}\right)\right)\right) \cdot \frac{1}{\cos \left(\frac{3}{2 \cdot \pi - \cos^{-1} \left(0 - \frac{g}{h}\right)} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9}\right)\right)}\right) \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(0 - \frac{g}{h}\right)\\ t_1 := \cos \left(\frac{\frac{{t\_0}^{2}}{9} + 0.4444444444444444 \cdot \left(\pi \cdot \pi\right)}{\pi \cdot 0.6666666666666666 - \frac{t\_0}{3}}\right)\\ \frac{2}{t\_1} \cdot \left(\cos \left(0.3333333333333333 \cdot \left(t\_0 + 2 \cdot \pi\right)\right) \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (- 0.0 (/ g h))))
        (t_1
         (cos
          (/
           (+ (/ (pow t_0 2.0) 9.0) (* 0.4444444444444444 (* PI PI)))
           (- (* PI 0.6666666666666666) (/ t_0 3.0))))))
   (* (/ 2.0 t_1) (* (cos (* 0.3333333333333333 (+ t_0 (* 2.0 PI)))) t_1))))

C

double code(double g, double h) {
	double t_0 = acos((0.0 - (g / h)));
	double t_1 = cos((((pow(t_0, 2.0) / 9.0) + (0.4444444444444444 * (((double) M_PI) * ((double) M_PI)))) / ((((double) M_PI) * 0.6666666666666666) - (t_0 / 3.0))));
	return (2.0 / t_1) * (cos((0.3333333333333333 * (t_0 + (2.0 * ((double) M_PI))))) * t_1);
}

Java

public static double code(double g, double h) {
	double t_0 = Math.acos((0.0 - (g / h)));
	double t_1 = Math.cos((((Math.pow(t_0, 2.0) / 9.0) + (0.4444444444444444 * (Math.PI * Math.PI))) / ((Math.PI * 0.6666666666666666) - (t_0 / 3.0))));
	return (2.0 / t_1) * (Math.cos((0.3333333333333333 * (t_0 + (2.0 * Math.PI)))) * t_1);
}

Python

def code(g, h):
	t_0 = math.acos((0.0 - (g / h)))
	t_1 = math.cos((((math.pow(t_0, 2.0) / 9.0) + (0.4444444444444444 * (math.pi * math.pi))) / ((math.pi * 0.6666666666666666) - (t_0 / 3.0))))
	return (2.0 / t_1) * (math.cos((0.3333333333333333 * (t_0 + (2.0 * math.pi)))) * t_1)

Julia

function code(g, h)
	t_0 = acos(Float64(0.0 - Float64(g / h)))
	t_1 = cos(Float64(Float64(Float64((t_0 ^ 2.0) / 9.0) + Float64(0.4444444444444444 * Float64(pi * pi))) / Float64(Float64(pi * 0.6666666666666666) - Float64(t_0 / 3.0))))
	return Float64(Float64(2.0 / t_1) * Float64(cos(Float64(0.3333333333333333 * Float64(t_0 + Float64(2.0 * pi)))) * t_1))
end

MATLAB

function tmp = code(g, h)
	t_0 = acos((0.0 - (g / h)));
	t_1 = cos(((((t_0 ^ 2.0) / 9.0) + (0.4444444444444444 * (pi * pi))) / ((pi * 0.6666666666666666) - (t_0 / 3.0))));
	tmp = (2.0 / t_1) * (cos((0.3333333333333333 * (t_0 + (2.0 * pi)))) * t_1);
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(0.0 - N[(g / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / 9.0), $MachinePrecision] + N[(0.4444444444444444 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.6666666666666666), $MachinePrecision] - N[(t$95$0 / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(2.0 / t$95$1), $MachinePrecision] * N[(N[Cos[N[(0.3333333333333333 * N[(t$95$0 + N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $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. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right), \color{blue}{2}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\cos \left(0.3333333333333333 \cdot \left(2 \cdot \pi + \cos^{-1} \left(0 - \frac{g}{h}\right)\right)\right) \cdot 2} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\left(\cos \left(0.3333333333333333 \cdot \left(\cos^{-1} \left(0 - \frac{g}{h}\right) + 2 \cdot \pi\right)\right) \cdot \cos \left(\frac{3}{2 \cdot \pi - \cos^{-1} \left(0 - \frac{g}{h}\right)} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9}\right)\right)\right) \cdot \frac{1}{\cos \left(\frac{3}{2 \cdot \pi - \cos^{-1} \left(0 - \frac{g}{h}\right)} \cdot \left(\pi \cdot \left(\pi \cdot 0.4444444444444444\right) + \frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9}\right)\right)}\right)} \cdot 2 \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{2}{\cos \left(\frac{\frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9} + \left(\pi \cdot \pi\right) \cdot 0.4444444444444444}{\pi \cdot 0.6666666666666666 - \frac{\cos^{-1} \left(0 - \frac{g}{h}\right)}{3}}\right)} \cdot \left(\cos \left(0.3333333333333333 \cdot \left(\pi \cdot 2 + \cos^{-1} \left(0 - \frac{g}{h}\right)\right)\right) \cdot \cos \left(\frac{\frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9} + \left(\pi \cdot \pi\right) \cdot 0.4444444444444444}{\pi \cdot 0.6666666666666666 - \frac{\cos^{-1} \left(0 - \frac{g}{h}\right)}{3}}\right)\right)} \]

9. Final simplification 100.0%

   \[\leadsto \frac{2}{\cos \left(\frac{\frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9} + 0.4444444444444444 \cdot \left(\pi \cdot \pi\right)}{\pi \cdot 0.6666666666666666 - \frac{\cos^{-1} \left(0 - \frac{g}{h}\right)}{3}}\right)} \cdot \left(\cos \left(0.3333333333333333 \cdot \left(\cos^{-1} \left(0 - \frac{g}{h}\right) + 2 \cdot \pi\right)\right) \cdot \cos \left(\frac{\frac{{\cos^{-1} \left(0 - \frac{g}{h}\right)}^{2}}{9} + 0.4444444444444444 \cdot \left(\pi \cdot \pi\right)}{\pi \cdot 0.6666666666666666 - \frac{\cos^{-1} \left(0 - \frac{g}{h}\right)}{3}}\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{\cos^{-1} \left(0 - \frac{g}{h}\right) \cdot -3 + \pi \cdot -6}{-9}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (/ (+ (* (acos (- 0.0 (/ g h))) -3.0) (* PI -6.0)) -9.0))))

C

double code(double g, double h) {
	return 2.0 * cos((((acos((0.0 - (g / h))) * -3.0) + (((double) M_PI) * -6.0)) / -9.0));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((Math.acos((0.0 - (g / h))) * -3.0) + (Math.PI * -6.0)) / -9.0));
}

Python

def code(g, h):
	return 2.0 * math.cos((((math.acos((0.0 - (g / h))) * -3.0) + (math.pi * -6.0)) / -9.0))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(acos(Float64(0.0 - Float64(g / h))) * -3.0) + Float64(pi * -6.0)) / -9.0)))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((acos((0.0 - (g / h))) * -3.0) + (pi * -6.0)) / -9.0));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(N[ArcCos[N[(0.0 - N[(g / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -3.0), $MachinePrecision] + N[(Pi * -6.0), $MachinePrecision]), $MachinePrecision] / -9.0), $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. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right)\right)\right) \]

   2. frac-2neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)}\right)\right)\right) \]

   3. frac-add N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}{3 \cdot \left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(3 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(0 - \frac{g}{h}\right) \cdot -3 + \pi \cdot -6}{-9}\right)} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \frac{\cos^{-1} \left(0 - \frac{g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (* PI 0.6666666666666666) (/ (acos (- 0.0 (/ g h))) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos(((((double) M_PI) * 0.6666666666666666) + (acos((0.0 - (g / h))) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos(((Math.PI * 0.6666666666666666) + (Math.acos((0.0 - (g / h))) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos(((math.pi * 0.6666666666666666) + (math.acos((0.0 - (g / h))) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(pi * 0.6666666666666666) + Float64(acos(Float64(0.0 - Float64(g / h))) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos(((pi * 0.6666666666666666) + (acos((0.0 - (g / h))) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(Pi * 0.6666666666666666), $MachinePrecision] + N[(N[ArcCos[N[(0.0 - N[(g / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 3.0), $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

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \frac{1}{3}\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot \frac{1}{3}\right)\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(2 \cdot \frac{1}{3}\right)\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(2 \cdot \frac{1}{3}\right)\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(2 \cdot \frac{1}{3}\right)\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

   7. metadata-eval 98.5%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{2}{3}\right), \mathsf{/.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(g\right), h\right)\right), 3\right)\right)\right)\right) \]

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \frac{\cos^{-1} \left(0 - \frac{g}{h}\right)}{3}\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))