2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 4.9s

Alternatives: 7

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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-acos.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

   10. add-sqr-sqrt N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

7. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(g, h)
	return Float64(sin(fma(-0.3333333333333333, fma(pi, 2.0, acos(Float64(Float64(-g) / h))), Float64(pi / 2.0))) * 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 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in g around 0

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

   1. distribute-rgt-in N/A

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

   2. mul-1-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-+l+ N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

8. Applied rewrites 99.9%

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

Alternative 5: 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

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

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 6: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. div-add-rev N/A

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

   6. frac-add N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. div-add N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

5. Applied rewrites 98.5%

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

Alternative 7: 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 2025101 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))