2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 10.2s

Precision: binary64

Cost: 59008

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\\ 2 \cdot \left(\sqrt[3]{{t_0}^{2}} \cdot \sqrt[3]{t_0}\right) \end{array} \]

FPCore

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

↓

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (cos (+ (* 0.6666666666666666 PI) (/ (acos (- (/ g h))) 3.0)))))
   (* 2.0 (* (cbrt (pow t_0 2.0)) (cbrt t_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) {
	double t_0 = cos(((0.6666666666666666 * ((double) M_PI)) + (acos(-(g / h)) / 3.0)));
	return 2.0 * (cbrt(pow(t_0, 2.0)) * cbrt(t_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) {
	double t_0 = Math.cos(((0.6666666666666666 * Math.PI) + (Math.acos(-(g / h)) / 3.0)));
	return 2.0 * (Math.cbrt(Math.pow(t_0, 2.0)) * Math.cbrt(t_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)
	t_0 = cos(Float64(Float64(0.6666666666666666 * pi) + Float64(acos(Float64(-Float64(g / h))) / 3.0)))
	return Float64(2.0 * Float64(cbrt((t_0 ^ 2.0)) * cbrt(t_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_] := Block[{t$95$0 = N[Cos[N[(N[(0.6666666666666666 * Pi), $MachinePrecision] + N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 1/3], $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. Applied egg-rr 1.0

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

4. Applied egg-rr 1.0

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	52928

\[\begin{array}{l} t_0 := 0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\\ 2 \cdot \left(\sqrt[3]{\cos t_0} \cdot \sqrt[3]{0.5 + 0.5 \cdot \cos \left(2 \cdot t_0\right)}\right) \end{array} \]

Alternative 2
Error	1.0
Cost	26176

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

Alternative 3
Error	1.0
Cost	19904

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

Reproduce

herbie shell --seed 2023187 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))