2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 4.0s

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ 2 \cdot \left(\cos \left(-0.3333333333333333 \cdot t\_0\right) \cdot -0.5 - \sin \left(0.3333333333333333 \cdot t\_0\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (*
    2.0
    (-
     (* (cos (* -0.3333333333333333 t_0)) -0.5)
     (*
      (sin (* 0.3333333333333333 t_0))
      (sin (- (* -0.6666666666666666 PI))))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	return 2.0 * ((cos((-0.3333333333333333 * t_0)) * -0.5) - (sin((0.3333333333333333 * t_0)) * sin(-(-0.6666666666666666 * ((double) M_PI)))));
}

Java

public static double code(double g, double h) {
	double t_0 = Math.acos((-g / h));
	return 2.0 * ((Math.cos((-0.3333333333333333 * t_0)) * -0.5) - (Math.sin((0.3333333333333333 * t_0)) * Math.sin(-(-0.6666666666666666 * Math.PI))));
}

Python

def code(g, h):
	t_0 = math.acos((-g / h))
	return 2.0 * ((math.cos((-0.3333333333333333 * t_0)) * -0.5) - (math.sin((0.3333333333333333 * t_0)) * math.sin(-(-0.6666666666666666 * math.pi))))

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	return Float64(2.0 * Float64(Float64(cos(Float64(-0.3333333333333333 * t_0)) * -0.5) - Float64(sin(Float64(0.3333333333333333 * t_0)) * sin(Float64(-Float64(-0.6666666666666666 * pi))))))
end

MATLAB

function tmp = code(g, h)
	t_0 = acos((-g / h));
	tmp = 2.0 * ((cos((-0.3333333333333333 * t_0)) * -0.5) - (sin((0.3333333333333333 * t_0)) * sin(-(-0.6666666666666666 * pi))));
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[(N[(N[Cos[N[(-0.3333333333333333 * t$95$0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[Sin[N[(0.3333333333333333 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[(-N[(-0.6666666666666666 * Pi), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right)} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 100.0

       \[\leadsto 2 \cdot \left(\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{-0.5} - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right) \]

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.5 - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right)} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (fma
    (sin (* t_0 -0.3333333333333333))
    (* (sin (* 0.3333333333333333 PI)) 2.0)
    (* -1.0 (cos (* 0.3333333333333333 t_0))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	return fma(sin((t_0 * -0.3333333333333333)), (sin((0.3333333333333333 * ((double) M_PI))) * 2.0), (-1.0 * cos((0.3333333333333333 * t_0))));
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	return fma(sin(Float64(t_0 * -0.3333333333333333)), Float64(sin(Float64(0.3333333333333333 * pi)) * 2.0), Float64(-1.0 * cos(Float64(0.3333333333333333 * t_0))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, N[(N[Sin[N[(t$95$0 * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(0.3333333333333333 * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] + N[(-1.0 * N[Cos[N[(0.3333333333333333 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right)} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 100.0

       \[\leadsto 2 \cdot \left(\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{-0.5} - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right) \]

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.5 - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(--0.6666666666666666 \cdot \pi\right)\right)} \]

7. Applied rewrites 100.0%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-acos.f64 N/A

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

   14. div-add-rev N/A

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

   15. lower-/.f64 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-flip N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. rem-square-sqrt N/A

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

   10. sqrt-unprod N/A

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

   11. rem-sqrt-square-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. fabs-mul N/A

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

   15. metadata-eval N/A

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

   16. associate-/l* N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. lift-/.f64 N/A

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

3. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025132 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))