2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 9.6s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{g}{-h}\right)\\ t_1 := \frac{\sqrt{3}}{2}\\ t_2 := 0.3333333333333333 \cdot t\_0\\ t_3 := \sin t\_2\\ t_4 := \left(0.25 - t\_1 \cdot t\_1\right) \cdot \cos t\_2\\ 2 \cdot \frac{{t\_4}^{3} - {\left(\mathsf{fma}\left(\cos \left(t\_0 \cdot 0.6666666666666666\right), -0.375, 0.375\right)\right)}^{1.5}}{{t\_4}^{2} + \left(0.5 \cdot \left(\sqrt{3} \cdot t\_3\right)\right) \cdot \mathsf{fma}\left(2, t\_3 \cdot \left(t\_1 \cdot 0.5\right), t\_4\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ g (- h))))
        (t_1 (/ (sqrt 3.0) 2.0))
        (t_2 (* 0.3333333333333333 t_0))
        (t_3 (sin t_2))
        (t_4 (* (- 0.25 (* t_1 t_1)) (cos t_2))))
   (*
    2.0
    (/
     (-
      (pow t_4 3.0)
      (pow (fma (cos (* t_0 0.6666666666666666)) -0.375 0.375) 1.5))
     (+
      (pow t_4 2.0)
      (* (* 0.5 (* (sqrt 3.0) t_3)) (fma 2.0 (* t_3 (* t_1 0.5)) t_4)))))))

C

double code(double g, double h) {
	double t_0 = acos((g / -h));
	double t_1 = sqrt(3.0) / 2.0;
	double t_2 = 0.3333333333333333 * t_0;
	double t_3 = sin(t_2);
	double t_4 = (0.25 - (t_1 * t_1)) * cos(t_2);
	return 2.0 * ((pow(t_4, 3.0) - pow(fma(cos((t_0 * 0.6666666666666666)), -0.375, 0.375), 1.5)) / (pow(t_4, 2.0) + ((0.5 * (sqrt(3.0) * t_3)) * fma(2.0, (t_3 * (t_1 * 0.5)), t_4))));
}

Julia

function code(g, h)
	t_0 = acos(Float64(g / Float64(-h)))
	t_1 = Float64(sqrt(3.0) / 2.0)
	t_2 = Float64(0.3333333333333333 * t_0)
	t_3 = sin(t_2)
	t_4 = Float64(Float64(0.25 - Float64(t_1 * t_1)) * cos(t_2))
	return Float64(2.0 * Float64(Float64((t_4 ^ 3.0) - (fma(cos(Float64(t_0 * 0.6666666666666666)), -0.375, 0.375) ^ 1.5)) / Float64((t_4 ^ 2.0) + Float64(Float64(0.5 * Float64(sqrt(3.0) * t_3)) * fma(2.0, Float64(t_3 * Float64(t_1 * 0.5)), t_4)))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[3.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] - N[Power[N[(N[Cos[N[(t$95$0 * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] * -0.375 + 0.375), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(N[(0.5 * N[(N[Sqrt[3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$3 * N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 99.9%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in g around 0

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

9. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - {\color{blue}{\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.6666666666666666\right), -0.375, 0.375\right)\right)}}^{1.5}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{2}{3}\right), \frac{-3}{8}, \frac{3}{8}\right)\right)}^{\frac{3}{2}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \color{blue}{\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)\right)} \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{2}{3}\right), \frac{-3}{8}, \frac{3}{8}\right)\right)}^{\frac{3}{2}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \color{blue}{\left(\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

    3. *-commutative N/A

       \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{2}{3}\right), \frac{-3}{8}, \frac{3}{8}\right)\right)}^{\frac{3}{2}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \color{blue}{\left(\sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right)\right)}\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

    4. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{2}{3}\right), \frac{-3}{8}, \frac{3}{8}\right)\right)}^{\frac{3}{2}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \color{blue}{\left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

    5. associate-*r* N/A

       \[\leadsto 2 \cdot \frac{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{2}{3}\right), \frac{-3}{8}, \frac{3}{8}\right)\right)}^{\frac{3}{2}}}{{\left(\left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)}^{2} + \left(2 \cdot \color{blue}{\left(\left(\sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \frac{\sqrt{3}}{2}\right) \cdot \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right), \left(\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

11. Applied rewrites 99.9%

    \[\leadsto 2 \cdot \frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.6666666666666666\right), -0.375, 0.375\right)\right)}^{1.5}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \color{blue}{\left(\left(\sqrt{3} \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right) \cdot 0.5\right)} \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)} \]

12. Final simplification 99.9%

    \[\leadsto 2 \cdot \frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)}^{3} - {\left(\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.6666666666666666\right), -0.375, 0.375\right)\right)}^{1.5}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)}^{2} + \left(0.5 \cdot \left(\sqrt{3} \cdot \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)\right) \cdot \mathsf{fma}\left(2, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)} \]
13. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(Pi * -0.1111111111111111), $MachinePrecision] * -6.0 + N[(0.3333333333333333 * N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. 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. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 98.5

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

6. Applied rewrites 98.5%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 98.5

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 98.5

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

   12. lift-/.f64 N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-/.f64 98.5

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. 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. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f64 N/A

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

   16. remove-double-neg N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 N/A

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

   19. metadata-eval 98.5

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

4. Applied rewrites 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%

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