2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 12.6s

Alternatives: 4

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{g}{-h}\right)\\ t_1 := 0.1111111111111111 \cdot {t\_0}^{2}\\ t_2 := \mathsf{fma}\left(0.3333333333333333, t\_0, -0.6666666666666666 \cdot \pi\right)\\ \mathsf{fma}\left(2, \sin \left(\frac{t\_1}{t\_2}\right) \cdot \sin \left(\frac{0.4444444444444444 \cdot \left(\pi \cdot \pi\right)}{t\_2}\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, 0.6666666666666666 \cdot \pi\right)\right) + \cos \left(\frac{\mathsf{fma}\left(\pi, 0.4444444444444444 \cdot \pi, t\_1\right)}{t\_2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ g (- h))))
        (t_1 (* 0.1111111111111111 (pow t_0 2.0)))
        (t_2 (fma 0.3333333333333333 t_0 (* -0.6666666666666666 PI))))
   (fma
    2.0
    (* (sin (/ t_1 t_2)) (sin (/ (* 0.4444444444444444 (* PI PI)) t_2)))
    (+
     (cos (fma 0.3333333333333333 t_0 (* 0.6666666666666666 PI)))
     (cos (/ (fma PI (* 0.4444444444444444 PI) t_1) t_2))))))

C

double code(double g, double h) {
	double t_0 = acos((g / -h));
	double t_1 = 0.1111111111111111 * pow(t_0, 2.0);
	double t_2 = fma(0.3333333333333333, t_0, (-0.6666666666666666 * ((double) M_PI)));
	return fma(2.0, (sin((t_1 / t_2)) * sin(((0.4444444444444444 * (((double) M_PI) * ((double) M_PI))) / t_2))), (cos(fma(0.3333333333333333, t_0, (0.6666666666666666 * ((double) M_PI)))) + cos((fma(((double) M_PI), (0.4444444444444444 * ((double) M_PI)), t_1) / t_2))));
}

Julia

function code(g, h)
	t_0 = acos(Float64(g / Float64(-h)))
	t_1 = Float64(0.1111111111111111 * (t_0 ^ 2.0))
	t_2 = fma(0.3333333333333333, t_0, Float64(-0.6666666666666666 * pi))
	return fma(2.0, Float64(sin(Float64(t_1 / t_2)) * sin(Float64(Float64(0.4444444444444444 * Float64(pi * pi)) / t_2))), Float64(cos(fma(0.3333333333333333, t_0, Float64(0.6666666666666666 * pi))) + cos(Float64(fma(pi, Float64(0.4444444444444444 * pi), t_1) / t_2))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.1111111111111111 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * t$95$0 + N[(-0.6666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[Sin[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(0.4444444444444444 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(0.3333333333333333 * t$95$0 + N[(0.6666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[(Pi * N[(0.4444444444444444 * Pi), $MachinePrecision] + t$95$1), $MachinePrecision] / t$95$2), $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. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 98.5

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f64 N/A

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

   16. remove-double-neg N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 N/A

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

   19. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. div-sub N/A

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

   7. cos-diff N/A

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

6. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\frac{{\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)}^{2}}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, -0.6666666666666666 \cdot \pi\right)}\right), \cos \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.4444444444444444}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, -0.6666666666666666 \cdot \pi\right)}\right), \sin \left(\frac{{\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)}^{2}}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, -0.6666666666666666 \cdot \pi\right)}\right) \cdot \sin \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.4444444444444444}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, -0.6666666666666666 \cdot \pi\right)}\right)\right)} \]

8. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{{\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)}^{2}}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot -0.6666666666666666\right)}\right), \sin \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.4444444444444444}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot -0.6666666666666666\right)}\right), \left(\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\right) + \cos \left(\frac{1}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), 0.3333333333333333, \pi \cdot -0.6666666666666666\right)} \cdot \mathsf{fma}\left(0.1111111111111111, {\cos^{-1} \left(\frac{g}{-h}\right)}^{2}, \left(\pi \cdot \pi\right) \cdot 0.4444444444444444\right)\right)\right) \cdot 0.5\right)} \]

9. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) + \cos \left(\frac{\frac{1}{9} \cdot {\cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}^{2} + \frac{4}{9} \cdot {\mathsf{PI}\left(\right)}^{2}}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}\right)\right) + \sin \left(\frac{1}{9} \cdot \frac{{\cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}^{2}}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}\right) \cdot \sin \left(\frac{4}{9} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}\right)\right)} \]

11. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sin \left(\frac{0.1111111111111111 \cdot {\cos^{-1} \left(-\frac{g}{h}\right)}^{2}}{\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(-\frac{g}{h}\right), \pi \cdot -0.6666666666666666\right)}\right) \cdot \sin \left(\frac{0.4444444444444444 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(-\frac{g}{h}\right), \pi \cdot -0.6666666666666666\right)}\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(-\frac{g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right) + \cos \left(\frac{\mathsf{fma}\left(\pi, \pi \cdot 0.4444444444444444, 0.1111111111111111 \cdot {\cos^{-1} \left(-\frac{g}{h}\right)}^{2}\right)}{\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(-\frac{g}{h}\right), \pi \cdot -0.6666666666666666\right)}\right)\right)} \]

12. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(2, \sin \left(\frac{0.1111111111111111 \cdot {\cos^{-1} \left(\frac{g}{-h}\right)}^{2}}{\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{g}{-h}\right), -0.6666666666666666 \cdot \pi\right)}\right) \cdot \sin \left(\frac{0.4444444444444444 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{g}{-h}\right), -0.6666666666666666 \cdot \pi\right)}\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{g}{-h}\right), 0.6666666666666666 \cdot \pi\right)\right) + \cos \left(\frac{\mathsf{fma}\left(\pi, 0.4444444444444444 \cdot \pi, 0.1111111111111111 \cdot {\cos^{-1} \left(\frac{g}{-h}\right)}^{2}\right)}{\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{g}{-h}\right), -0.6666666666666666 \cdot \pi\right)}\right)\right) \]
13. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), \frac{-1.5}{\pi}, -3\right)}{\frac{-1.5}{\pi} \cdot 3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (/ (fma (acos (/ g (- h))) (/ -1.5 PI) -3.0) (* (/ -1.5 PI) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((fma(acos((g / -h)), (-1.5 / ((double) M_PI)), -3.0) / ((-1.5 / ((double) M_PI)) * 3.0)));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(fma(acos(Float64(g / Float64(-h))), Float64(-1.5 / pi), -3.0) / Float64(Float64(-1.5 / pi) * 3.0))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * N[(-1.5 / Pi), $MachinePrecision] + -3.0), $MachinePrecision] / N[(N[(-1.5 / Pi), $MachinePrecision] * 3.0), $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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), \frac{-1.5}{\pi}, -3\right)}{3 \cdot \frac{-1.5}{\pi}}\right)} \]

5. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), \frac{-1.5}{\pi}, -3\right)}{\frac{-1.5}{\pi} \cdot 3}\right) \]
6. 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(g / Float64(-h))), -3.0, Float64(pi * -6.0)) / -9.0)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * -3.0 + N[(Pi * -6.0), $MachinePrecision]), $MachinePrecision] / -9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 98.5

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f64 N/A

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

   16. remove-double-neg N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 N/A

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

   19. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024222 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))