2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 4.2s

Alternatives: 6

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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ t_1 := \frac{t\_0}{3}\\ \cos t\_1 \cdot \cos \left(-0.6666666666666666 \cdot \pi\right) - \left(\sin t\_1 \cdot \sin \left(0.6666666666666666 \cdot \pi\right) - \cos \left(\frac{\mathsf{fma}\left(\pi, 2, t\_0\right)}{3}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))) (t_1 (/ t_0 3.0)))
   (-
    (* (cos t_1) (cos (* -0.6666666666666666 PI)))
    (-
     (* (sin t_1) (sin (* 0.6666666666666666 PI)))
     (cos (/ (fma PI 2.0 t_0) 3.0))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	double t_1 = t_0 / 3.0;
	return (cos(t_1) * cos((-0.6666666666666666 * ((double) M_PI)))) - ((sin(t_1) * sin((0.6666666666666666 * ((double) M_PI)))) - cos((fma(((double) M_PI), 2.0, t_0) / 3.0)));
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	t_1 = Float64(t_0 / 3.0)
	return Float64(Float64(cos(t_1) * cos(Float64(-0.6666666666666666 * pi))) - Float64(Float64(sin(t_1) * sin(Float64(0.6666666666666666 * pi))) - cos(Float64(fma(pi, 2.0, t_0) / 3.0))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 3.0), $MachinePrecision]}, N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[N[(-0.6666666666666666 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[N[(0.6666666666666666 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(N[(Pi * 2.0 + t$95$0), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ t_1 := 0.3333333333333333 \cdot t\_0\\ \cos t\_1 \cdot \cos \left(-0.6666666666666666 \cdot \pi\right) - \left(\left(0.5 \cdot \sin t\_1\right) \cdot \sqrt{3} - \cos \left(\frac{\mathsf{fma}\left(\pi, 2, t\_0\right)}{3}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))) (t_1 (* 0.3333333333333333 t_0)))
   (-
    (* (cos t_1) (cos (* -0.6666666666666666 PI)))
    (- (* (* 0.5 (sin t_1)) (sqrt 3.0)) (cos (/ (fma PI 2.0 t_0) 3.0))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	double t_1 = 0.3333333333333333 * t_0;
	return (cos(t_1) * cos((-0.6666666666666666 * ((double) M_PI)))) - (((0.5 * sin(t_1)) * sqrt(3.0)) - cos((fma(((double) M_PI), 2.0, t_0) / 3.0)));
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	t_1 = Float64(0.3333333333333333 * t_0)
	return Float64(Float64(cos(t_1) * cos(Float64(-0.6666666666666666 * pi))) - Float64(Float64(Float64(0.5 * sin(t_1)) * sqrt(3.0)) - cos(Float64(fma(pi, 2.0, t_0) / 3.0))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * t$95$0), $MachinePrecision]}, N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[N[(-0.6666666666666666 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(N[(Pi * 2.0 + t$95$0), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \pi\right) - \left(\sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right) - \cos \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{3}\right)\right)} \]
5. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. sin-2 N/A

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

   10. lower-*.f64 N/A

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

   11. cos-PI/3 N/A

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

   12. lower-*.f64 N/A

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

   13. sin-PI/3 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in g around 0

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

   1. lift-sqrt.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. mul-1-neg N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-acos.f64 N/A

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

   11. lower-*.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in g around 0

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

    1. mul-1-neg N/A

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

    2. distribute-frac-neg N/A

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

    3. lower-acos.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-neg.f64 N/A

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

    6. lift-*.f64 100.0

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

12. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\\ \frac{{\cos \left(t\_0 \cdot 0.3333333333333333\right)}^{3}}{\mathsf{fma}\left(\cos \left(t\_0 \cdot 0.6666666666666666\right), 0.5, 0.5\right)} \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (fma PI 2.0 (acos (/ (- g) h)))))
   (*
    (/
     (pow (cos (* t_0 0.3333333333333333)) 3.0)
     (fma (cos (* t_0 0.6666666666666666)) 0.5 0.5))
    2.0)))

C

double code(double g, double h) {
	double t_0 = fma(((double) M_PI), 2.0, acos((-g / h)));
	return (pow(cos((t_0 * 0.3333333333333333)), 3.0) / fma(cos((t_0 * 0.6666666666666666)), 0.5, 0.5)) * 2.0;
}

Julia

function code(g, h)
	t_0 = fma(pi, 2.0, acos(Float64(Float64(-g) / h)))
	return Float64(Float64((cos(Float64(t_0 * 0.3333333333333333)) ^ 3.0) / fma(cos(Float64(t_0 * 0.6666666666666666)), 0.5, 0.5)) * 2.0)
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Cos[N[(t$95$0 * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Cos[N[(t$95$0 * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $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. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.5

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in g around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.9%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 5: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. mul-1-neg N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. distribute-frac-neg N/A

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

   16. lower-acos.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. lift-neg.f64 N/A

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

   19. lower-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 6: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.5

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

5. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025072 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))