2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.6s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma
    (cbrt (* PI PI))
    (* (cbrt PI) 0.5)
    (fma -0.3333333333333333 (acos (/ (- g) h)) (* -0.6666666666666666 PI))))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(cbrt((((double) M_PI) * ((double) M_PI))), (cbrt(((double) M_PI)) * 0.5), fma(-0.3333333333333333, acos((-g / h)), (-0.6666666666666666 * ((double) M_PI)))));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(cbrt(Float64(pi * pi)), Float64(cbrt(pi) * 0.5), fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(-0.6666666666666666 * pi)))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * 0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(-0.6666666666666666 * Pi), $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. 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

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

3. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

3. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 99.9

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. common-denominator N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 98.5

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

3. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(Pi * 0.6666666666666666 + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. div-add N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. mult-flip N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lift-/.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 98.5

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

   18. lift-/.f64 N/A

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

   19. mult-flip N/A

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

3. Applied rewrites 98.5%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 1.1666666666666667 \cdot \pi\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin (fma (acos (/ (- g) h)) 0.3333333333333333 (* 1.1666666666666667 PI)))
  2.0))

C

double code(double g, double h) {
	return sin(fma(acos((-g / h)), 0.3333333333333333, (1.1666666666666667 * ((double) M_PI)))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(acos(Float64(Float64(-g) / h)), 0.3333333333333333, Float64(1.1666666666666667 * pi))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(1.1666666666666667 * 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. 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. count-2-rev N/A

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

   3. flip3-+ N/A

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

   4. lower-unsound-/.f64 N/A

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)}^{3} + {\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)}^{3}}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right), \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) - \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)\right)}} \]

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 1.1666666666666667 \cdot \pi\right)\right) \cdot 2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))