2-ancestry mixing, negative discriminant

Percentage Accurate: 98.4% → 100.0%

Time: 12.2s

Alternatives: 7

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: 98.4% 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

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

FPCore

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

C

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

Wolfram

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

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 99.9%

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

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]
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. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Simplified 100.0%

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

    1. lift-/.f64 N/A

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

    2. acos-neg N/A

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

    3. lift-PI.f64 N/A

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

    4. sub-neg N/A

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

    5. lift-PI.f64 N/A

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

    6. add-sqr-sqrt N/A

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

    7. lower-fma.f64 N/A

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

    8. lift-PI.f64 N/A

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

    9. lower-sqrt.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. lower-sqrt.f64 N/A

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

    12. acos-asin N/A

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

    13. sub-neg N/A

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

    14. asin-neg N/A

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

    15. lift-/.f64 N/A

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

    16. distribute-frac-neg2 N/A

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

    17. lift-neg.f64 N/A

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

    18. lift-/.f64 N/A

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

    19. lower-neg.f64 N/A

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

    20. lift-/.f64 N/A

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

    21. lift-neg.f64 N/A

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{g}{-h}\right)\\ -\mathsf{fma}\left(\sin \left(\left(t\_0 \cdot -3\right) \cdot -0.1111111111111111\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ g (- h)))))
   (-
    (fma
     (sin (* (* t_0 -3.0) -0.1111111111111111))
     (sqrt 3.0)
     (cos (* 0.3333333333333333 t_0))))))

C

double code(double g, double h) {
	double t_0 = acos((g / -h));
	return -fma(sin(((t_0 * -3.0) * -0.1111111111111111)), sqrt(3.0), cos((0.3333333333333333 * t_0)));
}

Julia

function code(g, h)
	t_0 = acos(Float64(g / Float64(-h)))
	return Float64(-fma(sin(Float64(Float64(t_0 * -3.0) * -0.1111111111111111)), sqrt(3.0), cos(Float64(0.3333333333333333 * t_0))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]}, (-N[(N[Sin[N[(N[(t$95$0 * -3.0), $MachinePrecision] * -0.1111111111111111), $MachinePrecision]], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[N[(0.3333333333333333 * t$95$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. 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 99.9%

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

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]
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. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Simplified 100.0%

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

    1. lift-/.f64 N/A

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

    2. lift-neg.f64 N/A

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

    3. lift-acos.f64 N/A

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. associate-*l* N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 100.0

       \[\leadsto -\mathsf{fma}\left(\sin \color{blue}{\left(\left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)}, \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \]

    9. lift-neg.f64 N/A

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

    10. lift-/.f64 N/A

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

    11. distribute-frac-neg2 N/A

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

    12. lift-neg.f64 N/A

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

    13. lift-/.f64 100.0

        \[\leadsto -\mathsf{fma}\left(\sin \left(\left(\cos^{-1} \color{blue}{\left(\frac{g}{-h}\right)} \cdot -3\right) \cdot -0.1111111111111111\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto -\mathsf{fma}\left(\sin \color{blue}{\left(\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)}, \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \]

12. Final simplification 100.0%

    \[\leadsto -\mathsf{fma}\left(\sin \left(\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right), \sqrt{3}, \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{g}{-h}\right)\\ -\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot t\_0\right), \sqrt{3}, \cos \left(\left(t\_0 \cdot -3\right) \cdot -0.1111111111111111\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ g (- h)))))
   (-
    (fma
     (sin (* 0.3333333333333333 t_0))
     (sqrt 3.0)
     (cos (* (* t_0 -3.0) -0.1111111111111111))))))

C

double code(double g, double h) {
	double t_0 = acos((g / -h));
	return -fma(sin((0.3333333333333333 * t_0)), sqrt(3.0), cos(((t_0 * -3.0) * -0.1111111111111111)));
}

Julia

function code(g, h)
	t_0 = acos(Float64(g / Float64(-h)))
	return Float64(-fma(sin(Float64(0.3333333333333333 * t_0)), sqrt(3.0), cos(Float64(Float64(t_0 * -3.0) * -0.1111111111111111))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]}, (-N[(N[Sin[N[(0.3333333333333333 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + N[Cos[N[(N[(t$95$0 * -3.0), $MachinePrecision] * -0.1111111111111111), $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 99.9%

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

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]
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. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Simplified 100.0%

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

    1. lift-/.f64 N/A

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

    2. lift-neg.f64 N/A

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

    3. lift-acos.f64 N/A

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. associate-*l* N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 100.0

       \[\leadsto -\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \color{blue}{\left(\left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)}\right) \]

    9. lift-neg.f64 N/A

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

    10. lift-/.f64 N/A

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

    11. distribute-frac-neg2 N/A

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

    12. lift-neg.f64 N/A

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

    13. lift-/.f64 100.0

        \[\leadsto -\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \left(\left(\cos^{-1} \color{blue}{\left(\frac{g}{-h}\right)} \cdot -3\right) \cdot -0.1111111111111111\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto -\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \sqrt{3}, \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)}\right) \]

12. Final simplification 100.0%

    \[\leadsto -\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right), \sqrt{3}, \cos \left(\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)\right) \]
13. Add Preprocessing

Alternative 4: 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 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
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 99.9%

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

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sqrt{3}\right)\right)} \]
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. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\mathsf{fma}\left(\pi \cdot -0.1111111111111111, -6, \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos
   (fma
    (* PI -0.1111111111111111)
    -6.0
    (* (* (acos (/ g (- h))) -3.0) -0.1111111111111111)))))

C

double code(double g, double h) {
	return 2.0 * cos(fma((((double) M_PI) * -0.1111111111111111), -6.0, ((acos((g / -h)) * -3.0) * -0.1111111111111111)));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(fma(Float64(pi * -0.1111111111111111), -6.0, Float64(Float64(acos(Float64(g / Float64(-h))) * -3.0) * -0.1111111111111111))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(Pi * -0.1111111111111111), $MachinePrecision] * -6.0 + N[(N[(N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision] * -3.0), $MachinePrecision] * -0.1111111111111111), $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. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. frac-2neg N/A

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

   12. frac-add N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

4. Applied egg-rr 98.5%

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

   1. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{g}{\color{blue}{\mathsf{neg}\left(h\right)}}\right) \cdot -3 + \mathsf{PI}\left(\right) \cdot -6}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \color{blue}{\left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3 + \mathsf{PI}\left(\right) \cdot -6}}\right) \]

   3. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3 + \mathsf{PI}\left(\right) \cdot -6}}\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3 + \color{blue}{\mathsf{PI}\left(\right)} \cdot -6}}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3 + \color{blue}{\mathsf{PI}\left(\right) \cdot -6}}}\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), -3, \mathsf{PI}\left(\right) \cdot -6\right)}}}\right) \]

   7. associate-/r/ N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{-9} \cdot \mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), -3, \mathsf{PI}\left(\right) \cdot -6\right)\right)} \]

   8. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3 + \mathsf{PI}\left(\right) \cdot -6\right)}\right) \]

   9. +-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6 + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right)}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{1}{-9} + \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{1}{-9}, \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \color{blue}{\frac{-1}{9}}, \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{-1}{9}, \color{blue}{\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}}\right)\right) \]

6. Applied egg-rr 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\pi \cdot -6, -0.1111111111111111, \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)\right)} \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot -6\right) \cdot \frac{-1}{9} + \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6\right)} \cdot \frac{-1}{9} + \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   3. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(\cos^{-1} \left(\frac{g}{\color{blue}{\mathsf{neg}\left(h\right)}}\right) \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(\cos^{-1} \color{blue}{\left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   5. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(\color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   6. associate-*l* N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \left(-3 \cdot \frac{-1}{9}\right)}\right) \]

   7. metadata-eval N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \color{blue}{\frac{1}{3}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}}\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{-1}{9} \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{-1}{9} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6\right)} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \]

   11. associate-*r* N/A

       \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   13. lower-*.f64 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{-0.1111111111111111 \cdot \pi}, -6, \cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right) \]

8. Applied egg-rr 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(-0.1111111111111111 \cdot \pi, -6, \cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{g}{h}}\right)\right) \cdot \frac{1}{3}\right)\right) \]

   2. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)} \cdot \frac{1}{3}\right)\right) \]

   3. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \color{blue}{\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)} \cdot \frac{1}{3}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \color{blue}{\left(-3 \cdot \frac{-1}{9}\right)}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \color{blue}{\left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot -3\right) \cdot \frac{-1}{9}}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \color{blue}{\left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot -3\right)} \cdot \frac{-1}{9}\right)\right) \]

   7. lift-*.f64 98.5

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(-0.1111111111111111 \cdot \pi, -6, \color{blue}{\left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot -3\right) \cdot -0.1111111111111111}\right)\right) \]

   8. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \left(\cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)} \cdot -3\right) \cdot \frac{-1}{9}\right)\right) \]

   9. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \left(\cos^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{g}{h}}\right)\right) \cdot -3\right) \cdot \frac{-1}{9}\right)\right) \]

   10. distribute-frac-neg2 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \left(\cos^{-1} \color{blue}{\left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3\right) \cdot \frac{-1}{9}\right)\right) \]

   11. lift-neg.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \left(\cos^{-1} \left(\frac{g}{\color{blue}{\mathsf{neg}\left(h\right)}}\right) \cdot -3\right) \cdot \frac{-1}{9}\right)\right) \]

   12. lift-/.f64 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(-0.1111111111111111 \cdot \pi, -6, \left(\cos^{-1} \color{blue}{\left(\frac{g}{-h}\right)} \cdot -3\right) \cdot -0.1111111111111111\right)\right) \]

10. Applied egg-rr 98.5%

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(-0.1111111111111111 \cdot \pi, -6, \color{blue}{\left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111}\right)\right) \]

11. Final simplification 98.5%

    \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi \cdot -0.1111111111111111, -6, \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)\right) \]
12. Add Preprocessing

Alternative 6: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\mathsf{fma}\left(\pi \cdot -0.1111111111111111, -6, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos
   (fma
    (* PI -0.1111111111111111)
    -6.0
    (* 0.3333333333333333 (acos (/ g (- h))))))))

C

double code(double g, double h) {
	return 2.0 * cos(fma((((double) M_PI) * -0.1111111111111111), -6.0, (0.3333333333333333 * acos((g / -h)))));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(fma(Float64(pi * -0.1111111111111111), -6.0, Float64(0.3333333333333333 * acos(Float64(g / Float64(-h)))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(Pi * -0.1111111111111111), $MachinePrecision] * -6.0 + N[(0.3333333333333333 * N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. 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. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. frac-2neg N/A

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

   12. frac-add N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

4. Applied egg-rr 98.5%

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

   1. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{g}{\color{blue}{\mathsf{neg}\left(h\right)}}\right) \cdot -3 + \mathsf{PI}\left(\right) \cdot -6}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \color{blue}{\left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3 + \mathsf{PI}\left(\right) \cdot -6}}\right) \]

   3. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3 + \mathsf{PI}\left(\right) \cdot -6}}\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3 + \color{blue}{\mathsf{PI}\left(\right)} \cdot -6}}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3 + \color{blue}{\mathsf{PI}\left(\right) \cdot -6}}}\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), -3, \mathsf{PI}\left(\right) \cdot -6\right)}}}\right) \]

   7. associate-/r/ N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{-9} \cdot \mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), -3, \mathsf{PI}\left(\right) \cdot -6\right)\right)} \]

   8. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3 + \mathsf{PI}\left(\right) \cdot -6\right)}\right) \]

   9. +-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6 + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right)}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{1}{-9} + \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{1}{-9}, \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \color{blue}{\frac{-1}{9}}, \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{-1}{9}, \color{blue}{\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{1}{-9}}\right)\right) \]

6. Applied egg-rr 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\pi \cdot -6, -0.1111111111111111, \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot -3\right) \cdot -0.1111111111111111\right)\right)} \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot -6\right) \cdot \frac{-1}{9} + \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6\right)} \cdot \frac{-1}{9} + \left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   3. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(\cos^{-1} \left(\frac{g}{\color{blue}{\mathsf{neg}\left(h\right)}}\right) \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(\cos^{-1} \color{blue}{\left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   5. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(\color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot -3\right) \cdot \frac{-1}{9}\right) \]

   6. associate-*l* N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \left(-3 \cdot \frac{-1}{9}\right)}\right) \]

   7. metadata-eval N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \color{blue}{\frac{1}{3}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}}\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{-1}{9} \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{-1}{9} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6\right)} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \]

   11. associate-*r* N/A

       \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6} + \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{1}{3}\right)\right)} \]

   13. lower-*.f64 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{-0.1111111111111111 \cdot \pi}, -6, \cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right) \]

8. Applied egg-rr 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(-0.1111111111111111 \cdot \pi, -6, \cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)} \]

9. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi \cdot -0.1111111111111111, -6, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{-h}\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 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 98.5%

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

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. frac-2neg N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. frac-2neg N/A

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

   11. div-inv N/A

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

   12. distribute-rgt-out N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 98.5

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

   17. lift-/.f64 N/A

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

   18. frac-2neg N/A

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

4. Applied egg-rr 98.5%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))