2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 2.8s

Alternatives: 4

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (/ (fma PI 2.0 (acos (/ (- g) h))) 3.0)) (t_1 (cos t_0)))
   (/
    (* (pow t_1 3.0) 2.0)
    (fma t_1 t_1 (- (pow t_1 2.0) (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))))))))

C

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

Julia

function code(g, h)
	t_0 = Float64(fma(pi, 2.0, acos(Float64(Float64(-g) / h))) / 3.0)
	t_1 = cos(t_0)
	return Float64(Float64((t_1 ^ 3.0) * 2.0) / fma(t_1, t_1, Float64((t_1 ^ 2.0) - Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))))))
end

Wolfram

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

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. 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-/.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 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. sqr-cos-a N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. count-2-rev N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in g around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-acos.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-*.f64 98.5

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

6. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

8. Applied rewrites 99.9%

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

Alternative 3: 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. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)} \]
3. 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 98.4

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

4. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right)} \]
5. 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-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   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. mul-1-neg N/A

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

   14. distribute-frac-neg N/A

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

   15. lower-acos.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. lower-*.f64 98.5

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

6. Applied rewrites 98.5%

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

Alternative 4: 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. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)} \]
3. 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 98.4

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

4. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))