2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 10.9s

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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\\ \left(1.2307692307692308 \cdot \left(-0.8125 \cdot \sin t\_0\right)\right) \cdot \sqrt{3} - \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (- (/ g h))))))
   (- (* (* 1.2307692307692308 (* -0.8125 (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 ((1.2307692307692308 * (-0.8125 * sin(t_0))) * sqrt(3.0)) - cos(t_0);
}

Fortran

real(8) function code(g, h)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8) :: t_0
    t_0 = 0.3333333333333333d0 * acos(-(g / h))
    code = ((1.2307692307692308d0 * ((-0.8125d0) * sin(t_0))) * sqrt(3.0d0)) - cos(t_0)
end function

Java

public static double code(double g, double h) {
	double t_0 = 0.3333333333333333 * Math.acos(-(g / h));
	return ((1.2307692307692308 * (-0.8125 * Math.sin(t_0))) * Math.sqrt(3.0)) - Math.cos(t_0);
}

Python

def code(g, h):
	t_0 = 0.3333333333333333 * math.acos(-(g / h))
	return ((1.2307692307692308 * (-0.8125 * math.sin(t_0))) * math.sqrt(3.0)) - math.cos(t_0)

Julia

function code(g, h)
	t_0 = Float64(0.3333333333333333 * acos(Float64(-Float64(g / h))))
	return Float64(Float64(Float64(1.2307692307692308 * Float64(-0.8125 * sin(t_0))) * sqrt(3.0)) - cos(t_0))
end

MATLAB

function tmp = code(g, h)
	t_0 = 0.3333333333333333 * acos(-(g / h));
	tmp = ((1.2307692307692308 * (-0.8125 * 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[(N[(1.2307692307692308 * N[(-0.8125 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision] - N[Cos[t$95$0], $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%

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

6. Applied rewrites 100.0%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-acos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\left(1.2307692307692308 \cdot \left(-0.8125 \cdot \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)\right) \cdot \sqrt{3}} + \left(-\cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(1.2307692307692308 \cdot \left(-0.8125 \cdot \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right)\right) \cdot \sqrt{3} - \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot -0.40625, 2, 0.8125 \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\sqrt{3} \cdot -0.5\right) \cdot 2\right)\right)\right)}{1.625}} \]

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{\frac{16}{13} \cdot \left(\frac{-13}{16} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-13}{16} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]
8. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{16}{13} \cdot \color{blue}{\left(\frac{-13}{16} \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{16}{13} \cdot \frac{-13}{16}\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)} \]

   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. lower-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. lower-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. Applied rewrites 99.9%

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

11. Applied rewrites 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 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 98.5%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot -0.40625, 2, 0.8125 \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\sqrt{3} \cdot -0.5\right) \cdot 2\right)\right)\right)}{1.625}} \]

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{\frac{16}{13} \cdot \left(\frac{-13}{16} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-13}{16} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]
8. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{16}{13} \cdot \color{blue}{\left(\frac{-13}{16} \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{16}{13} \cdot \frac{-13}{16}\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)} \]

   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. lower-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. lower-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. Applied rewrites 99.9%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right)} \]
10. 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(-Float64(g / 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-PI.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.5

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

   14. lift-/.f64 N/A

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

   15. div-inv N/A

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

   16. lower-*.f64 N/A

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

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