2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 2.9s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-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. lower-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-PI.f64 97.5

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

3. Applied rewrites 97.5%

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

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}, \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{-3}\right), \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(-\pi\right) \cdot 0.6666666666666666\right)\right)} \]

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[Sin[N[(Pi * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] + (-N[Cos[t$95$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-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. lower-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-PI.f64 97.5

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

3. Applied rewrites 97.5%

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

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}, \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{-3}\right), \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(-\pi\right) \cdot 0.6666666666666666\right)\right)} \]

6. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{-1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{4} \cdot {\left(\sqrt{3}\right)}^{2}\right) + \sin \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)} \]

8. Applied rewrites 98.5%

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

9. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{-1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{4} \cdot {\left(\sqrt{3}\right)}^{2}\right) + \sin \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)\right)} \]

11. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-add-rev N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-PI.f64 98.5

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

3. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $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. 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. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-acos.f64 N/A

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

   6. lower-fma.f64 N/A

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

   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. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. lift-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 98.5%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(Pi * 1.1666666666666667), $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. 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. lower-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-PI.f64 97.5

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

3. Applied rewrites 97.5%

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

4. Taylor expanded in g around 0

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

6. Applied rewrites 97.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. metadata-eval 97.6

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

8. Applied rewrites 97.6%

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

Reproduce

herbie shell --seed 2025140 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))