2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.3s

Alternatives: 3

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, 1.0× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin (/ (- (* (+ (+ PI (acos (/ (- g) h))) PI) 2.0) (* 3.0 PI)) -6.0))))

C

double code(double g, double h) {
	return 2.0 * sin((((((((double) M_PI) + acos((-g / h))) + ((double) M_PI)) * 2.0) - (3.0 * ((double) M_PI))) / -6.0));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.sin((((((Math.PI + Math.acos((-g / h))) + Math.PI) * 2.0) - (3.0 * Math.PI)) / -6.0));
}

Python

def code(g, h):
	return 2.0 * math.sin((((((math.pi + math.acos((-g / h))) + math.pi) * 2.0) - (3.0 * math.pi)) / -6.0))

Julia

function code(g, h)
	return Float64(2.0 * sin(Float64(Float64(Float64(Float64(Float64(pi + acos(Float64(Float64(-g) / h))) + pi) * 2.0) - Float64(3.0 * pi)) / -6.0)))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * sin((((((pi + acos((-g / h))) + pi) * 2.0) - (3.0 * pi)) / -6.0));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(N[(N[(N[(Pi + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision] * 2.0), $MachinePrecision] - N[(3.0 * Pi), $MachinePrecision]), $MachinePrecision] / -6.0), $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
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-add N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. metadata-eval N/A

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

   6. count-2-rev N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-PI.f64 100.0

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

   12. lift-PI.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-PI.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot -0.16666666666666666 - 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (- (* PI -0.16666666666666666) (* 0.3333333333333333 (acos (/ (- g) h)))))
  2.0))

C

double code(double g, double h) {
	return sin(((((double) M_PI) * -0.16666666666666666) - (0.3333333333333333 * acos((-g / h))))) * 2.0;
}

Java

public static double code(double g, double h) {
	return Math.sin(((Math.PI * -0.16666666666666666) - (0.3333333333333333 * Math.acos((-g / h))))) * 2.0;
}

Python

def code(g, h):
	return math.sin(((math.pi * -0.16666666666666666) - (0.3333333333333333 * math.acos((-g / h))))) * 2.0

Julia

function code(g, h)
	return Float64(sin(Float64(Float64(pi * -0.16666666666666666) - Float64(0.3333333333333333 * acos(Float64(Float64(-g) / h))))) * 2.0)
end

MATLAB

function tmp = code(g, h)
	tmp = sin(((pi * -0.16666666666666666) - (0.3333333333333333 * acos((-g / h))))) * 2.0;
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[(Pi * -0.16666666666666666), $MachinePrecision] - N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in g around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. distribute-rgt-in N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-acos.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate--l+ N/A

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

   11. lower-+.f64 N/A

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

7. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

9. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot -0.16666666666666666 - 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2} \]
10. Add Preprocessing

Alternative 3: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double g, double h) {
	return Math.cos((-0.3333333333333333 * ((Math.PI + Math.acos((-g / h))) + Math.PI))) * 2.0;
}

Python

def code(g, h):
	return math.cos((-0.3333333333333333 * ((math.pi + math.acos((-g / h))) + math.pi))) * 2.0

Julia

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

MATLAB

function tmp = code(g, h)
	tmp = cos((-0.3333333333333333 * ((pi + acos((-g / h))) + pi))) * 2.0;
end

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-acos.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 98.5

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

   11. lift-+.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. fp-cancel-sign-sub-inv N/A

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

   15. distribute-lft-neg-in N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 98.5%

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

   1. lift--.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. metadata-eval N/A

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

   6. count-2-rev N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-PI.f64 98.5

       \[\leadsto \cos \left(-0.3333333333333333 \cdot \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) + \pi\right) + \pi\right)\right) \cdot 2 \]

   12. lift-PI.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-PI.f64 98.5

       \[\leadsto \cos \left(-0.3333333333333333 \cdot \left(\left(\pi + \cos^{-1} \left(\frac{-g}{h}\right)\right) + \pi\right)\right) \cdot 2 \]

9. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025058 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))