2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 10.4s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

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

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

   6. +-commutative N/A

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

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

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

9. Simplified 100.0%

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

    1. *-commutative N/A

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

    2. rem-square-sqrt N/A

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

    3. unpow2 N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

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

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

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

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

    10. neg-lowering-neg.f64 100.0

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

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

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

   6. +-commutative N/A

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

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

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 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(Float64(-g) / 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 98.4%

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

   1. +-commutative N/A

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

   2. frac-2neg N/A

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

   3. frac-add N/A

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

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

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

4. Applied egg-rr 98.4%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative N/A

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

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

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))