2-ancestry mixing, negative discriminant

Average Error: 1.0 → 1.0

Time: 3.3s

Precision: binary64

Cost: 19904

Math

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

↓

\[2 \cdot \cos \left(0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) \]

FPCore

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

↓

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (* 0.6666666666666666 PI) (/ (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)));
}

↓

double code(double g, double h) {
	return 2.0 * cos(((0.6666666666666666 * ((double) M_PI)) + (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)));
}

↓

public static double code(double g, double h) {
	return 2.0 * Math.cos(((0.6666666666666666 * Math.PI) + (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)))

↓

def code(g, h):
	return 2.0 * math.cos(((0.6666666666666666 * math.pi) + (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

↓

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(0.6666666666666666 * pi) + 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

↓

function tmp = code(g, h)
	tmp = 2.0 * cos(((0.6666666666666666 * pi) + (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]

↓

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(0.6666666666666666 * Pi), $MachinePrecision] + N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.0

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

2. Simplified 1.0

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

   [Start] 1.0	\[ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
   associate-/l* [=>] 1.0	\[ 2 \cdot \cos \left(\color{blue}{\frac{2}{\frac{3}{\pi}}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
   associate-/r/ [=>] 1.0	\[ 2 \cdot \cos \left(\color{blue}{\frac{2}{3} \cdot \pi} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
   metadata-eval [=>] 1.0	\[ 2 \cdot \cos \left(\color{blue}{0.6666666666666666} \cdot \pi + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]

3. Final simplification 1.0

   \[\leadsto 2 \cdot \cos \left(0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) \]

Alternatives

Alternative 1
Error	2.2
Cost	19840

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

Reproduce

herbie shell --seed 2023028 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))