2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 8.2s

Alternatives: 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 100.0%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 100.0

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

8. Applied rewrites 100.0%

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

Reproduce

herbie shell --seed 2025059 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))