2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 3.8s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

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

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

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. count-2 N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-lft1-in N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\\ \frac{4 \cdot {\cos t\_0}^{3}}{\cos \left(t\_0 \cdot 2\right) - -1} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0
         (fma
          0.3333333333333333
          (acos (/ (- g) h))
          (* PI 0.6666666666666666))))
   (/ (* 4.0 (pow (cos t_0) 3.0)) (- (cos (* t_0 2.0)) -1.0))))

C

double code(double g, double h) {
	double t_0 = fma(0.3333333333333333, acos((-g / h)), (((double) M_PI) * 0.6666666666666666));
	return (4.0 * pow(cos(t_0), 3.0)) / (cos((t_0 * 2.0)) - -1.0);
}

Julia

function code(g, h)
	t_0 = fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(pi * 0.6666666666666666))
	return Float64(Float64(4.0 * (cos(t_0) ^ 3.0)) / Float64(cos(Float64(t_0 * 2.0)) - -1.0))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(N[(4.0 * N[Power[N[Cos[t$95$0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

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

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. count-2 N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-lft1-in N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

9. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{4 \cdot {\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right)}^{3}}{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right) \cdot 2\right) - -1}} \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

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 4: 98.5% accurate, 1.1× speedup

Math

\[2 \cdot \cos \left(\left(6.283185307179587 + \left(\pi - \cos^{-1} \left(\frac{g}{h}\right)\right)\right) \cdot 0.3333333333333333\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (* (+ 6.283185307179587 (- PI (acos (/ g h)))) 0.3333333333333333))))

C

double code(double g, double h) {
	return 2.0 * cos(((6.283185307179587 + (((double) M_PI) - acos((g / h)))) * 0.3333333333333333));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos(((6.283185307179587 + (Math.PI - Math.acos((g / h)))) * 0.3333333333333333));
}

Python

def code(g, h):
	return 2.0 * math.cos(((6.283185307179587 + (math.pi - math.acos((g / h)))) * 0.3333333333333333))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(6.283185307179587 + Float64(pi - acos(Float64(g / h)))) * 0.3333333333333333)))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos(((6.283185307179587 + (pi - acos((g / h)))) * 0.3333333333333333));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(6.283185307179587 + N[(Pi - N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \color{blue}{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}}\right) \]

   3. add-to-fraction N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   4. mult-flip N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \]

   5. metadata-eval N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{\frac{1}{3}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \]

   7. lower-+.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \frac{1}{3}\right) \]

   8. metadata-eval 98.5%

      \[\leadsto 2 \cdot \cos \left(\left(\color{blue}{6.283185307179587} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(6.283185307179587 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right)}\right) \cdot \frac{1}{3}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \color{blue}{\left(\frac{-g}{h}\right)}\right) \cdot \frac{1}{3}\right) \]

   3. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right)\right) \cdot \frac{1}{3}\right) \]

   4. distribute-frac-neg N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}\right) \cdot \frac{1}{3}\right) \]

   5. acos-neg N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \color{blue}{\left(\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)\right)}\right) \cdot \frac{1}{3}\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \left(\color{blue}{\pi} - \cos^{-1} \left(\frac{g}{h}\right)\right)\right) \cdot \frac{1}{3}\right) \]

   7. lower--.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \color{blue}{\left(\pi - \cos^{-1} \left(\frac{g}{h}\right)\right)}\right) \cdot \frac{1}{3}\right) \]

   8. lower-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{7074237752028441}{1125899906842624} + \left(\pi - \color{blue}{\cos^{-1} \left(\frac{g}{h}\right)}\right)\right) \cdot \frac{1}{3}\right) \]

   9. lower-/.f64 98.5%

      \[\leadsto 2 \cdot \cos \left(\left(6.283185307179587 + \left(\pi - \cos^{-1} \color{blue}{\left(\frac{g}{h}\right)}\right)\right) \cdot 0.3333333333333333\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \left(\left(6.283185307179587 + \color{blue}{\left(\pi - \cos^{-1} \left(\frac{g}{h}\right)\right)}\right) \cdot 0.3333333333333333\right) \]
7. Add Preprocessing

Alternative 5: 98.5% accurate, 1.1× speedup

Math

\[2 \cdot \cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot 0.3333333333333333\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (* (- (acos (/ (- g) h)) -6.283185307179587) 0.3333333333333333))))

C

double code(double g, double h) {
	return 2.0 * cos(((acos((-g / h)) - -6.283185307179587) * 0.3333333333333333));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    code = 2.0d0 * cos(((acos((-g / h)) - (-6.283185307179587d0)) * 0.3333333333333333d0))
end function

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos(((Math.acos((-g / h)) - -6.283185307179587) * 0.3333333333333333));
}

Python

def code(g, h):
	return 2.0 * math.cos(((math.acos((-g / h)) - -6.283185307179587) * 0.3333333333333333))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(acos(Float64(Float64(-g) / h)) - -6.283185307179587) * 0.3333333333333333)))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos(((acos((-g / h)) - -6.283185307179587) * 0.3333333333333333));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] - -6.283185307179587), $MachinePrecision] * 0.3333333333333333), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \color{blue}{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}}\right) \]

   3. add-to-fraction N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   4. mult-flip N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \]

   5. metadata-eval N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{\frac{1}{3}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \]

   7. lower-+.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{2358079250676147}{1125899906842624} \cdot 3 + \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \frac{1}{3}\right) \]

   8. metadata-eval 98.5%

      \[\leadsto 2 \cdot \cos \left(\left(\color{blue}{6.283185307179587} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(6.283185307179587 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \frac{1}{3}\right) \]

   2. +-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) + \frac{7074237752028441}{1125899906842624}\right)} \cdot \frac{1}{3}\right) \]

   3. add-flip N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)\right)} \cdot \frac{1}{3}\right) \]

   4. lower--.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)\right)} \cdot \frac{1}{3}\right) \]

   5. metadata-eval 98.5%

      \[\leadsto 2 \cdot \cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{-6.283185307179587}\right) \cdot 0.3333333333333333\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right)} \cdot 0.3333333333333333\right) \]
7. Add Preprocessing

Alternative 6: 98.4% accurate, 1.1× speedup

Math

\[\cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right) - 2.0943951023931957\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (- (* -0.3333333333333333 (acos (/ (- g) h))) 2.0943951023931957))
  2.0))

C

double code(double g, double h) {
	return cos(((-0.3333333333333333 * acos((-g / h))) - 2.0943951023931957)) * 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    code = cos((((-0.3333333333333333d0) * acos((-g / h))) - 2.0943951023931957d0)) * 2.0d0
end function

Java

public static double code(double g, double h) {
	return Math.cos(((-0.3333333333333333 * Math.acos((-g / h))) - 2.0943951023931957)) * 2.0;
}

Python

def code(g, h):
	return math.cos(((-0.3333333333333333 * math.acos((-g / h))) - 2.0943951023931957)) * 2.0

Julia

function code(g, h)
	return Float64(cos(Float64(Float64(-0.3333333333333333 * acos(Float64(Float64(-g) / h))) - 2.0943951023931957)) * 2.0)
end

MATLAB

function tmp = code(g, h)
	tmp = cos(((-0.3333333333333333 * acos((-g / h))) - 2.0943951023931957)) * 2.0;
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 2.0943951023931957), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.4%

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{-2358079250676147}{1125899906842624}\right)} \cdot 2 \]

   2. add-flip N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-2358079250676147}{1125899906842624}\right)\right)\right)} \cdot 2 \]

   3. lower--.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-2358079250676147}{1125899906842624}\right)\right)\right)} \cdot 2 \]

   4. lower-*.f64 N/A

      \[\leadsto \cos \left(\color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} - \left(\mathsf{neg}\left(\frac{-2358079250676147}{1125899906842624}\right)\right)\right) \cdot 2 \]

   5. metadata-eval 98.4%

      \[\leadsto \cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{2.0943951023931957}\right) \cdot 2 \]

6. Applied rewrites 98.4%

   \[\leadsto \cos \color{blue}{\left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right) - 2.0943951023931957\right)} \cdot 2 \]
7. Add Preprocessing

Alternative 7: 98.4% accurate, 1.2× speedup

Math

\[\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma -0.3333333333333333 (acos (/ (- g) h)) -2.0943951023931957))
  2.0))

C

double code(double g, double h) {
	return cos(fma(-0.3333333333333333, acos((-g / h)), -2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), -2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + -2.0943951023931957), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.4%

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 1.2× speedup

Math

\[\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 3.6651914291880923\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (sin (fma (acos (/ (- g) h)) 0.3333333333333333 3.6651914291880923)) 2.0))

C

double code(double g, double h) {
	return sin(fma(acos((-g / h)), 0.3333333333333333, 3.6651914291880923)) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(acos(Float64(Float64(-g) / h)), 0.3333333333333333, 3.6651914291880923)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + 3.6651914291880923), $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. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

3. Applied rewrites 97.5%

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

4. Evaluated real constant 97.6%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{3.6651914291880923}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{4126638688683257}{1125899906842624}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{4126638688683257}{1125899906842624}\right)\right) \cdot 2} \]

   3. lower-*.f64 97.6%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{4126638688683257}{1125899906842624}\right)} \cdot 2 \]

   5. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{1}{3}} + \frac{4126638688683257}{1125899906842624}\right) \cdot 2 \]

   6. lower-fma.f64 97.6%

      \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 3.6651914291880923\right)\right)} \cdot 2 \]

6. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 3.6651914291880923\right)\right) \cdot 2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))