2-ancestry mixing, negative discriminant

Percentage Accurate: 98.4% → 100.0%

Time: 11.6s

Alternatives: 6

Speedup: 1.0×

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: 98.4% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\\ \mathsf{fma}\left(\sqrt{3} \cdot -0.8125, \sin t\_0, -0.8125 \cdot \cos t\_0\right) \cdot 1.2307692307692308 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (/ g (- h))))))
   (*
    (fma (* (sqrt 3.0) -0.8125) (sin t_0) (* -0.8125 (cos t_0)))
    1.2307692307692308)))

C

double code(double g, double h) {
	double t_0 = 0.3333333333333333 * acos((g / -h));
	return fma((sqrt(3.0) * -0.8125), sin(t_0), (-0.8125 * cos(t_0))) * 1.2307692307692308;
}

Julia

function code(g, h)
	t_0 = Float64(0.3333333333333333 * acos(Float64(g / Float64(-h))))
	return Float64(fma(Float64(sqrt(3.0) * -0.8125), sin(t_0), Float64(-0.8125 * cos(t_0))) * 1.2307692307692308)
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sqrt[3.0], $MachinePrecision] * -0.8125), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision] + N[(-0.8125 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.2307692307692308), $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

4. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}, \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)} \]

6. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot -0.40625, 2, 0.8125 \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\sqrt{3} \cdot -0.5\right) \cdot 2\right)\right)\right)}{1.625}} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \frac{-13}{32}\right) \cdot 2 + \frac{13}{16} \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\sqrt{3} \cdot \frac{-1}{2}\right) \cdot 2\right)\right)}}{\frac{13}{8}} \]

   2. +-commutative N/A

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{13}{16} \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\sqrt{3} \cdot \frac{-1}{2}\right) \cdot 2\right)\right) + \left(\cos \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \cdot \frac{-13}{32}\right) \cdot 2}}{\frac{13}{8}} \]

8. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.8125 \cdot \left(-\sqrt{3}\right), \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.8125\right)}}{1.625} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{\mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right)}{\frac{13}{8}}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right)}{\frac{13}{8}} \cdot 2} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right)}{\frac{13}{8}}} \cdot 2 \]

   4. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right) \cdot \frac{1}{\frac{13}{8}}\right)} \cdot 2 \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right) \cdot \left(\frac{1}{\frac{13}{8}} \cdot 2\right)} \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right) \cdot \left(\color{blue}{\frac{8}{13}} \cdot 2\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{13}{16} \cdot \left(\mathsf{neg}\left(\sqrt{3}\right)\right), \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right), \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)\right) \cdot \frac{-13}{16}\right) \cdot \color{blue}{\frac{16}{13}} \]

   8. lower-*.f64 100.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.8125 \cdot \left(-\sqrt{3}\right), \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.8125\right) \cdot 1.2307692307692308} \]

10. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{3} \cdot -0.8125, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot -0.8125\right) \cdot 1.2307692307692308} \]

11. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\sqrt{3} \cdot -0.8125, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), -0.8125 \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right) \cdot 1.2307692307692308 \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\\ 1.2307692307692308 \cdot \left(-0.8125 \cdot \mathsf{fma}\left(\sin t\_0, \sqrt{3}, \cos t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (/ g (- h))))))
   (* 1.2307692307692308 (* -0.8125 (fma (sin t_0) (sqrt 3.0) (cos t_0))))))

C

double code(double g, double h) {
	double t_0 = 0.3333333333333333 * acos((g / -h));
	return 1.2307692307692308 * (-0.8125 * fma(sin(t_0), sqrt(3.0), cos(t_0)));
}

Julia

function code(g, h)
	t_0 = Float64(0.3333333333333333 * acos(Float64(g / Float64(-h))))
	return Float64(1.2307692307692308 * Float64(-0.8125 * fma(sin(t_0), sqrt(3.0), cos(t_0))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(1.2307692307692308 * N[(-0.8125 * N[(N[Sin[t$95$0], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[t$95$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

4. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}, \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)} \]

6. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot -0.40625, 2, 0.8125 \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\sqrt{3} \cdot -0.5\right) \cdot 2\right)\right)\right)}{1.625}} \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), -0.8125, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot \left(\left(-\sqrt{3}\right) \cdot 0.8125\right)\right) \cdot 1.2307692307692308} \]

9. Taylor expanded in g around 0

   \[\leadsto \color{blue}{\left(\frac{-13}{16} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-13}{16} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \cdot \frac{16}{13} \]
10. Step-by-step derivation

    1. distribute-lft-out N/A

       \[\leadsto \color{blue}{\left(\frac{-13}{16} \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \cdot \frac{16}{13} \]

    2. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{-13}{16} \cdot \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \cdot \frac{16}{13} \]

    3. +-commutative N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \color{blue}{\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3} + \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)}\right) \cdot \frac{16}{13} \]

    4. lower-fma.f64 N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)}\right) \cdot \frac{16}{13} \]

    5. lower-sin.f64 N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)}, \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)}, \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    7. lower-acos.f64 N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    8. mul-1-neg N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    9. lower-neg.f64 N/A

       \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    10. lower-/.f64 N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{g}{h}}\right)\right)\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    11. lower-sqrt.f64 N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right), \color{blue}{\sqrt{3}}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)\right) \cdot \frac{16}{13} \]

    12. lower-cos.f64 N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right), \sqrt{3}, \color{blue}{\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)}\right)\right) \cdot \frac{16}{13} \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right), \sqrt{3}, \cos \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)}\right)\right) \cdot \frac{16}{13} \]

    14. lower-acos.f64 N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(-1 \cdot \frac{g}{h}\right)}\right)\right)\right) \cdot \frac{16}{13} \]

    15. mul-1-neg N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}\right)\right)\right) \cdot \frac{16}{13} \]

    16. lower-neg.f64 N/A

        \[\leadsto \left(\frac{-13}{16} \cdot \mathsf{fma}\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right), \sqrt{3}, \cos \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}\right)\right)\right) \cdot \frac{16}{13} \]

    17. lower-/.f64 100.0

        \[\leadsto \left(-0.8125 \cdot \mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\color{blue}{\frac{g}{h}}\right)\right)\right)\right) \cdot 1.2307692307692308 \]

11. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\left(-0.8125 \cdot \mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right)\right)} \cdot 1.2307692307692308 \]

12. Final simplification 100.0%

    \[\leadsto 1.2307692307692308 \cdot \left(-0.8125 \cdot \mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (/ g (- h))))))
   (- (fma (sin t_0) (sqrt 3.0) (cos t_0)))))

C

double code(double g, double h) {
	double t_0 = 0.3333333333333333 * acos((g / -h));
	return -fma(sin(t_0), sqrt(3.0), cos(t_0));
}

Julia

function code(g, h)
	t_0 = Float64(0.3333333333333333 * acos(Float64(g / Float64(-h))))
	return Float64(-fma(sin(t_0), sqrt(3.0), cos(t_0)))
end

Wolfram

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

4. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}, \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)} \]

5. Taylor expanded in g around 0

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-out N/A

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

   10. associate-*r* N/A

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

7. Applied rewrites 100.0%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), -3, \pi \cdot -6\right)}}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (/ 1.0 (/ -9.0 (fma (acos (/ g (- h))) -3.0 (* PI -6.0)))))))

C

double code(double g, double h) {
	return 2.0 * cos((1.0 / (-9.0 / fma(acos((g / -h)), -3.0, (((double) M_PI) * -6.0)))));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(1.0 / Float64(-9.0 / fma(acos(Float64(g / Float64(-h))), -3.0, Float64(pi * -6.0))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(1.0 / N[(-9.0 / N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * -3.0 + N[(Pi * -6.0), $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. +-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. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. distribute-rgt-neg-out N/A

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

   13. *-commutative N/A

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

   14. distribute-lft-neg-out N/A

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

   15. lower-fma.f64 N/A

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

4. Applied rewrites 98.4%

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

Alternative 5: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 6: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(0.3333333333333333 * N[(2.0 * Pi + 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

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

   5. div-inv N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 98.4

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. lift-neg.f64 N/A

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

   14. remove-double-neg N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 98.4

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

4. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))