2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 3.8s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

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

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

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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := {t\_0}^{3}\\ \frac{t\_2 + t\_2}{\mathsf{fma}\left(t\_0, t\_0, t\_1 - t\_1\right)} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0
         (cos
          (fma
           0.3333333333333333
           (acos (/ (- g) h))
           (* PI 0.6666666666666666))))
        (t_1 (* t_0 t_0))
        (t_2 (pow t_0 3.0)))
   (/ (+ t_2 t_2) (fma t_0 t_0 (- t_1 t_1)))))

C

double code(double g, double h) {
	double t_0 = cos(fma(0.3333333333333333, acos((-g / h)), (((double) M_PI) * 0.6666666666666666)));
	double t_1 = t_0 * t_0;
	double t_2 = pow(t_0, 3.0);
	return (t_2 + t_2) / fma(t_0, t_0, (t_1 - t_1));
}

Julia

function code(g, h)
	t_0 = cos(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(pi * 0.6666666666666666)))
	t_1 = Float64(t_0 * t_0)
	t_2 = t_0 ^ 3.0
	return Float64(Float64(t_2 + t_2) / fma(t_0, t_0, Float64(t_1 - t_1)))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[Cos[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 3.0], $MachinePrecision]}, N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(t$95$1 - t$95$1), $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

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

   2. count-2-rev N/A

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

   3. flip3-+ N/A

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

   4. lower-unsound-/.f64 N/A

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)}^{3} + {\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)}^{3}}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) - \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)\right)}} \]
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\\ {\cos t\_0}^{3} \cdot \frac{-4}{-1 - \cos \left(t\_0 \cdot 2\right)} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0
         (fma
          0.3333333333333333
          (acos (/ (- g) h))
          (* 0.6666666666666666 PI))))
   (* (pow (cos t_0) 3.0) (/ -4.0 (- -1.0 (cos (* t_0 2.0)))))))

C

double code(double g, double h) {
	double t_0 = fma(0.3333333333333333, acos((-g / h)), (0.6666666666666666 * ((double) M_PI)));
	return pow(cos(t_0), 3.0) * (-4.0 / (-1.0 - cos((t_0 * 2.0))));
}

Julia

function code(g, h)
	t_0 = fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(0.6666666666666666 * pi))
	return Float64((cos(t_0) ^ 3.0) * Float64(-4.0 / Float64(-1.0 - cos(Float64(t_0 * 2.0)))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(0.6666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 3.0], $MachinePrecision] * N[(-4.0 / N[(-1.0 - N[Cos[N[(t$95$0 * 2.0), $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. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. div-add N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. mult-flip N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lift-/.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 98.5

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

   18. lift-/.f64 N/A

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

   19. mult-flip N/A

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)}^{3} \cdot 2}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right) \cdot 2\right), 0.5, 0.5\right)}} \]

7. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

9. Applied rewrites 99.9%

   \[\leadsto \color{blue}{{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right)}^{3} \cdot \frac{-4}{-1 - \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right) \cdot 2\right)}} \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

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. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. div-add N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. mult-flip N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lift-/.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 98.5

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

   18. lift-/.f64 N/A

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

   19. mult-flip N/A

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

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

\[\cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot -0.3333333333333333\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (* (- (acos (/ (- g) h)) -6.283185307179587) -0.3333333333333333))
  2.0))

C

double code(double g, double h) {
	return cos(((acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    code = cos(((acos((-g / h)) - (-6.283185307179587d0)) * (-0.3333333333333333d0))) * 2.0d0
end function

Java

public static double code(double g, double h) {
	return Math.cos(((Math.acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0;
}

Python

def code(g, h):
	return math.cos(((math.acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0

Julia

function code(g, h)
	return Float64(cos(Float64(Float64(acos(Float64(Float64(-g) / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0)
end

MATLAB

function tmp = code(g, h)
	tmp = cos(((acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0;
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] - -6.283185307179587), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 2.0), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2358079250676147}{1125899906842624}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.4

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{-2358079250676147}{1125899906842624}\right)} \cdot 2 \]

   2. +-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-2358079250676147}{1125899906842624} + \frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot 2 \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-2358079250676147}{1125899906842624} - \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot 2 \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\frac{-2358079250676147}{1125899906842624} - \color{blue}{\frac{1}{3}} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2 \]

   5. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{-2358079250676147}{1125899906842624} - \color{blue}{\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right) \cdot 2 \]

   6. sub-negate-rev N/A

      \[\leadsto \cos \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-2358079250676147}{1125899906842624}\right)\right)\right)} \cdot 2 \]

   7. metadata-eval N/A

      \[\leadsto \cos \left(\mathsf{neg}\left(\left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{2358079250676147}{1125899906842624}\right)\right)}\right)\right)\right) \cdot 2 \]

   8. add-flip N/A

      \[\leadsto \cos \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{2358079250676147}{1125899906842624}\right)}\right)\right) \cdot 2 \]

   9. lift-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{neg}\left(\left(\color{blue}{\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} + \frac{2358079250676147}{1125899906842624}\right)\right)\right) \cdot 2 \]

   10. metadata-eval N/A

       \[\leadsto \cos \left(\mathsf{neg}\left(\left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \color{blue}{\frac{1}{3} \cdot \frac{7074237752028441}{1125899906842624}}\right)\right)\right) \cdot 2 \]

   11. distribute-lft-in N/A

       \[\leadsto \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) + \frac{7074237752028441}{1125899906842624}\right)}\right)\right) \cdot 2 \]

   12. +-commutative N/A

       \[\leadsto \cos \left(\mathsf{neg}\left(\frac{1}{3} \cdot \color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right)\right) \cdot 2 \]

   13. lift-+.f64 N/A

       \[\leadsto \cos \left(\mathsf{neg}\left(\frac{1}{3} \cdot \color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right)\right) \cdot 2 \]

   14. *-commutative N/A

       \[\leadsto \cos \left(\mathsf{neg}\left(\color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}}\right)\right) \cdot 2 \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \cos \color{blue}{\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} \cdot 2 \]

   16. metadata-eval N/A

       \[\leadsto \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{\frac{-1}{3}}\right) \cdot 2 \]

   17. lower-*.f64 98.5

       \[\leadsto \cos \color{blue}{\left(\left(6.283185307179587 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.3333333333333333\right)} \cdot 2 \]

6. Applied rewrites 98.5%

   \[\leadsto \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot -0.3333333333333333\right)} \cdot 2 \]
7. Add Preprocessing

Alternative 5: 98.4% accurate, 1.2× speedup

Math

\[\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma -0.3333333333333333 (acos (/ (- g) h)) -2.0943951023931957))
  2.0))

C

double code(double g, double h) {
	return cos(fma(-0.3333333333333333, acos((-g / h)), -2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), -2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + -2.0943951023931957), $MachinePrecision]], $MachinePrecision] * 2.0), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2358079250676147}{1125899906842624}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.4

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025180 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))