2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 9.7s

Alternatives: 4

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(-0.5 \cdot \mathsf{fma}\left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi}, -\cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (*
   -0.5
   (fma
    (sin (* (acos (/ g (- h))) 0.3333333333333333))
    (sqrt 3.0)
    (cos
     (* 0.3333333333333333 (fma (sqrt PI) (sqrt PI) (- (acos (/ g h))))))))))

C

double code(double g, double h) {
	return 2.0 * (-0.5 * fma(sin((acos((g / -h)) * 0.3333333333333333)), sqrt(3.0), cos((0.3333333333333333 * fma(sqrt(((double) M_PI)), sqrt(((double) M_PI)), -acos((g / h)))))));
}

Julia

function code(g, h)
	return Float64(2.0 * Float64(-0.5 * fma(sin(Float64(acos(Float64(g / Float64(-h))) * 0.3333333333333333)), sqrt(3.0), cos(Float64(0.3333333333333333 * fma(sqrt(pi), sqrt(pi), Float64(-acos(Float64(g / h)))))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[(-0.5 * N[(N[Sin[N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[N[(0.3333333333333333 * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision] + (-N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision])), $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. Add Preprocessing

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. mul-1-neg N/A

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

   12. lower-/.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-cos.f64 N/A

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

9. Simplified 99.9%

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

    1. distribute-frac-neg2 N/A

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

    2. acos-neg N/A

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

    3. lift-PI.f64 N/A

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

    4. sub-neg N/A

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

    5. rem-square-sqrt N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. acos-asin N/A

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

    9. unsub-neg N/A

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

    10. asin-neg N/A

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

    11. distribute-frac-neg2 N/A

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

    12. lift-neg.f64 N/A

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

    13. lift-/.f64 N/A

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

    14. lower-fma.f64 N/A

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

    15. lower-neg.f64 N/A

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

    16. lift-/.f64 N/A

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

    17. lift-neg.f64 N/A

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

    18. distribute-frac-neg2 N/A

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(-0.5 \cdot \mathsf{fma}\left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \left(\pi - \cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (*
   -0.5
   (fma
    (sin (* (acos (/ g (- h))) 0.3333333333333333))
    (sqrt 3.0)
    (cos (* 0.3333333333333333 (- PI (acos (/ g h)))))))))

C

double code(double g, double h) {
	return 2.0 * (-0.5 * fma(sin((acos((g / -h)) * 0.3333333333333333)), sqrt(3.0), cos((0.3333333333333333 * (((double) M_PI) - acos((g / h)))))));
}

Julia

function code(g, h)
	return Float64(2.0 * Float64(-0.5 * fma(sin(Float64(acos(Float64(g / Float64(-h))) * 0.3333333333333333)), sqrt(3.0), cos(Float64(0.3333333333333333 * Float64(pi - acos(Float64(g / h))))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[(-0.5 * N[(N[Sin[N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[N[(0.3333333333333333 * N[(Pi - N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision]), $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. Add Preprocessing

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. mul-1-neg N/A

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

   12. lower-/.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-cos.f64 N/A

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

9. Simplified 99.9%

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

    1. distribute-frac-neg2 N/A

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

    2. acos-neg N/A

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

    3. lift-PI.f64 N/A

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

    4. acos-asin N/A

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

    5. unsub-neg N/A

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

    6. asin-neg N/A

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

    7. distribute-frac-neg2 N/A

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

    8. lift-neg.f64 N/A

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

    9. lift-/.f64 N/A

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

    10. lower--.f64 N/A

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

    11. lift-/.f64 N/A

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

    12. lift-neg.f64 N/A

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

    13. distribute-frac-neg2 N/A

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

    14. asin-neg N/A

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

    15. unsub-neg N/A

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

    16. acos-asin N/A

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

    17. lower-acos.f64 N/A

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

    18. lower-/.f64 100.0

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, N[(2.0 * N[(-0.5 * 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 98.5%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in g around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. mul-1-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. mul-1-neg N/A

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

   12. lower-/.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-cos.f64 N/A

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

9. Simplified 99.9%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(Pi * 0.6666666666666666 + N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.5

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

   14. lift-/.f64 N/A

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

   15. div-inv N/A

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

   16. lower-*.f64 N/A

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

4. Applied egg-rr 98.5%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))