2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 3.3s

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, 0.5× 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

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

   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. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

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

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(4.0 / Float64(cos(Float64(t_0 * 2.0)) - -1.0)))
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[(4.0 / N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] - -1.0), $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

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

   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

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

9. Applied rewrites 99.9%

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

Alternative 3: 98.5% accurate, 1.2× speedup

Math

\[\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -0.523598775598299\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (sin (fma (acos (/ (- g) h)) -0.3333333333333333 -0.523598775598299)) 2.0))

C

double code(double g, double h) {
	return sin(fma(acos((-g / h)), -0.3333333333333333, -0.523598775598299)) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(acos(Float64(Float64(-g) / h)), -0.3333333333333333, -0.523598775598299)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333 + -0.523598775598299), $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 98.4%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-PI.f64 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 97.6%

       \[\leadsto \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \color{blue}{\mathsf{fma}\left(0.5, \pi, -2.0943951023931957\right)}\right)\right) \cdot 2 \]

6. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \mathsf{fma}\left(0.5, \pi, -2.0943951023931957\right)\right)\right)} \cdot 2 \]

7. Evaluated real constant 98.5%

   \[\leadsto \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \color{blue}{-0.523598775598299}\right)\right) \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025185 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))