2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 9.5s

Alternatives: 4

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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ t_1 := 0.5 \cdot \sqrt{3}\\ t_2 := \frac{\sqrt{3}}{2}\\ t_3 := \left(0.25 - t\_2 \cdot t\_2\right) \cdot \cos \left(t\_0 \cdot -0.3333333333333333\right)\\ t_4 := 0.3333333333333333 \cdot t\_0\\ t_5 := \sin t\_4\\ \frac{{t\_3}^{3} - {\left(0.75 \cdot \left(0.5 - \cos \left(t\_4 \cdot 2\right) \cdot 0.5\right)\right)}^{1.5}}{\mathsf{fma}\left(t\_5, t\_1, t\_3\right) \cdot \left(t\_1 \cdot t\_5\right) + {t\_3}^{2}} \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h)))
        (t_1 (* 0.5 (sqrt 3.0)))
        (t_2 (/ (sqrt 3.0) 2.0))
        (t_3 (* (- 0.25 (* t_2 t_2)) (cos (* t_0 -0.3333333333333333))))
        (t_4 (* 0.3333333333333333 t_0))
        (t_5 (sin t_4)))
   (*
    (/
     (- (pow t_3 3.0) (pow (* 0.75 (- 0.5 (* (cos (* t_4 2.0)) 0.5))) 1.5))
     (+ (* (fma t_5 t_1 t_3) (* t_1 t_5)) (pow t_3 2.0)))
    2.0)))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	double t_1 = 0.5 * sqrt(3.0);
	double t_2 = sqrt(3.0) / 2.0;
	double t_3 = (0.25 - (t_2 * t_2)) * cos((t_0 * -0.3333333333333333));
	double t_4 = 0.3333333333333333 * t_0;
	double t_5 = sin(t_4);
	return ((pow(t_3, 3.0) - pow((0.75 * (0.5 - (cos((t_4 * 2.0)) * 0.5))), 1.5)) / ((fma(t_5, t_1, t_3) * (t_1 * t_5)) + pow(t_3, 2.0))) * 2.0;
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	t_1 = Float64(0.5 * sqrt(3.0))
	t_2 = Float64(sqrt(3.0) / 2.0)
	t_3 = Float64(Float64(0.25 - Float64(t_2 * t_2)) * cos(Float64(t_0 * -0.3333333333333333)))
	t_4 = Float64(0.3333333333333333 * t_0)
	t_5 = sin(t_4)
	return Float64(Float64(Float64((t_3 ^ 3.0) - (Float64(0.75 * Float64(0.5 - Float64(cos(Float64(t_4 * 2.0)) * 0.5))) ^ 1.5)) / Float64(Float64(fma(t_5, t_1, t_3) * Float64(t_1 * t_5)) + (t_3 ^ 2.0))) * 2.0)
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[3.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.25 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(t$95$0 * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.3333333333333333 * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$4], $MachinePrecision]}, N[(N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] - N[Power[N[(0.75 * N[(0.5 - N[(N[Cos[N[(t$95$4 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$5 * t$95$1 + t$95$3), $MachinePrecision] * N[(t$95$1 * t$95$5), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$3, 2.0], $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

4. Applied rewrites 99.9%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity 99.9

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

8. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 3.0 + N[(Pi * 6.0), $MachinePrecision]), $MachinePrecision] / 9.0), $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. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, \color{blue}{6} \cdot \mathsf{PI}\left(\right)\right)}{3 \cdot 3}\right) \]

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, \color{blue}{6} \cdot \mathsf{PI}\left(\right)\right)}{3 \cdot 3}\right) \]

   16. metadata-eval 98.5

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \pi\right)}{\color{blue}{9}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \pi\right)}{9}\right)} \]

5. Final simplification 98.5%

   \[\leadsto \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, \pi \cdot 6\right)}{9}\right) \cdot 2 \]
6. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(0.1111111111111111 \cdot \pi, 6, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos
   (fma
    (* 0.1111111111111111 PI)
    6.0
    (* 0.3333333333333333 (acos (/ (- g) h)))))
  2.0))

C

double code(double g, double h) {
	return cos(fma((0.1111111111111111 * ((double) M_PI)), 6.0, (0.3333333333333333 * acos((-g / h))))) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(Float64(0.1111111111111111 * pi), 6.0, Float64(0.3333333333333333 * acos(Float64(Float64(-g) / h))))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(0.1111111111111111 * Pi), $MachinePrecision] * 6.0 + N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $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. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, \color{blue}{6} \cdot \mathsf{PI}\left(\right)\right)}{3 \cdot 3}\right) \]

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, \color{blue}{6} \cdot \mathsf{PI}\left(\right)\right)}{3 \cdot 3}\right) \]

   16. metadata-eval 98.5

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \pi\right)}{\color{blue}{9}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \pi\right)}{9}\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}{9}\right)} \]

   2. clear-num N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{\frac{9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]

   3. associate-/r/ N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{9} \cdot \mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{9} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3 + 6 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   5. +-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{9} \cdot \color{blue}{\left(6 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right)}\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{9} + \left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(6 \cdot \mathsf{PI}\left(\right), \frac{1}{9}, \left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{6 \cdot \mathsf{PI}\left(\right)}, \frac{1}{9}, \left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot 6}, \frac{1}{9}, \left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot 6}, \frac{1}{9}, \left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 6, \color{blue}{\frac{1}{9}}, \left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 6, \frac{1}{9}, \color{blue}{\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot 3\right) \cdot \frac{1}{9}}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 6, \frac{1}{9}, \color{blue}{\left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)} \cdot \frac{1}{9}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 6, \frac{1}{9}, \color{blue}{\left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)} \cdot \frac{1}{9}\right)\right) \]

   15. metadata-eval 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi \cdot 6, 0.1111111111111111, \left(3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{0.1111111111111111}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\pi \cdot 6, 0.1111111111111111, \left(3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.1111111111111111\right)\right)} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot \frac{1}{9} + \left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{9}\right)} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{1}{9} \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)} + \left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{9}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{9} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 6\right)} + \left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{9}\right) \]

   4. associate-*r* N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot 6} + \left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{9}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right), 6, \left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{9}\right)\right)} \]

   6. lower-*.f64 98.5

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{0.1111111111111111 \cdot \pi}, 6, \left(3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.1111111111111111\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right), 6, \color{blue}{\left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{9}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right), 6, \color{blue}{\frac{1}{9} \cdot \left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right), 6, \frac{1}{9} \cdot \color{blue}{\left(3 \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)}\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right), 6, \color{blue}{\left(\frac{1}{9} \cdot 3\right) \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{9} \cdot \mathsf{PI}\left(\right), 6, \color{blue}{\frac{1}{3}} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right) \]

   12. lift-*.f64 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(0.1111111111111111 \cdot \pi, 6, \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right)\right) \]

8. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.1111111111111111 \cdot \pi, 6, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)} \]

9. Final simplification 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.1111111111111111 \cdot \pi, 6, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \cdot 2 \]
10. Add Preprocessing

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   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. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 98.5

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))