2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 11.9s

Alternatives: 4

Speedup: 0.5×

Specification

Math

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

FPCore

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(-\frac{g}{h}\right)\\ t_1 := 0.3333333333333333 \cdot t\_0\\ \sin t\_1 \cdot \left(-\sqrt{3}\right) - \cos \left(\sqrt{0.3333333333333333 \cdot t\_1} \cdot \sqrt{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (- (/ g h)))) (t_1 (* 0.3333333333333333 t_0)))
   (-
    (* (sin t_1) (- (sqrt 3.0)))
    (cos (* (sqrt (* 0.3333333333333333 t_1)) (sqrt t_0))))))

C

double code(double g, double h) {
	double t_0 = acos(-(g / h));
	double t_1 = 0.3333333333333333 * t_0;
	return (sin(t_1) * -sqrt(3.0)) - cos((sqrt((0.3333333333333333 * t_1)) * sqrt(t_0)));
}

Fortran

real(8) function code(g, h)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: t_1
    t_0 = acos(-(g / h))
    t_1 = 0.3333333333333333d0 * t_0
    code = (sin(t_1) * -sqrt(3.0d0)) - cos((sqrt((0.3333333333333333d0 * t_1)) * sqrt(t_0)))
end function

Java

public static double code(double g, double h) {
	double t_0 = Math.acos(-(g / h));
	double t_1 = 0.3333333333333333 * t_0;
	return (Math.sin(t_1) * -Math.sqrt(3.0)) - Math.cos((Math.sqrt((0.3333333333333333 * t_1)) * Math.sqrt(t_0)));
}

Python

def code(g, h):
	t_0 = math.acos(-(g / h))
	t_1 = 0.3333333333333333 * t_0
	return (math.sin(t_1) * -math.sqrt(3.0)) - math.cos((math.sqrt((0.3333333333333333 * t_1)) * math.sqrt(t_0)))

Julia

function code(g, h)
	t_0 = acos(Float64(-Float64(g / h)))
	t_1 = Float64(0.3333333333333333 * t_0)
	return Float64(Float64(sin(t_1) * Float64(-sqrt(3.0))) - cos(Float64(sqrt(Float64(0.3333333333333333 * t_1)) * sqrt(t_0))))
end

MATLAB

function tmp = code(g, h)
	t_0 = acos(-(g / h));
	t_1 = 0.3333333333333333 * t_0;
	tmp = (sin(t_1) * -sqrt(3.0)) - cos((sqrt((0.3333333333333333 * t_1)) * sqrt(t_0)));
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * t$95$0), $MachinePrecision]}, N[(N[(N[Sin[t$95$1], $MachinePrecision] * (-N[Sqrt[3.0], $MachinePrecision])), $MachinePrecision] - N[Cos[N[(N[Sqrt[N[(0.3333333333333333 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 99.9%

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

8. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-*.f64 99.9

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqrt-prod N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (- (/ g h))))))
   (- (* (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 (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 = (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 (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 (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(sin(t_0) * Float64(-sqrt(3.0))) - cos(t_0))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 99.9%

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

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right) \cdot \left(-\sqrt{3}\right) - \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)} \]
9. 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%

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in g around 0

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

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

   3. frac-2neg N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. frac-2neg N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. frac-2neg N/A

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

   11. div-inv N/A

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

   12. distribute-rgt-out N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 98.5

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

   17. lift-/.f64 N/A

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

   18. frac-2neg N/A

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))