2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 4.2s

Alternatives: 3

Speedup: 1.2×

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, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, \pi \cdot 0.6666666666666666\right)\\ {\cos t\_0}^{3} \cdot \frac{2}{\mathsf{fma}\left(\cos \left(t\_0 \cdot -2\right), 0.5, 0.5\right)} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0
         (fma
          (acos (/ (- g) h))
          0.3333333333333333
          (* PI 0.6666666666666666))))
   (* (pow (cos t_0) 3.0) (/ 2.0 (fma (cos (* t_0 -2.0)) 0.5 0.5)))))

C

double code(double g, double h) {
	double t_0 = fma(acos((-g / h)), 0.3333333333333333, (((double) M_PI) * 0.6666666666666666));
	return pow(cos(t_0), 3.0) * (2.0 / fma(cos((t_0 * -2.0)), 0.5, 0.5));
}

Julia

function code(g, h)
	t_0 = fma(acos(Float64(Float64(-g) / h)), 0.3333333333333333, Float64(pi * 0.6666666666666666))
	return Float64((cos(t_0) ^ 3.0) * Float64(2.0 / fma(cos(Float64(t_0 * -2.0)), 0.5, 0.5)))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(Pi * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 / N[(N[Cos[N[(t$95$0 * -2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $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. *-commutative N/A

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

   3. lower-*.f64 98.5%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

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) / (cos((t_0 * 2.0)) - -1.0)) * 4.0;
}

Julia

function code(g, h)
	t_0 = fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(0.6666666666666666 * pi))
	return Float64(Float64((cos(t_0) ^ 3.0) / Float64(cos(Float64(t_0 * 2.0)) - -1.0)) * 4.0)
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[(N[Power[N[Cos[t$95$0], $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 4.0), $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. 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. *-commutative N/A

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

   3. lower-*.f64 98.5%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-lft1-in N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

9. Applied rewrites 99.9%

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

Alternative 3: 98.5% 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%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
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 98.5%

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

Reproduce

herbie shell --seed 2025193 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))