Average Error: 1.0 → 1.0
Time: 2.4s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
↓
\[2 \cdot \cos \left(\mathsf{fma}\left(2, \pi, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot 0.3333333333333333\right) \]
(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 (* (fma 2.0 PI (acos (/ g (- h)))) 0.3333333333333333))))
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((fma(2.0, ((double) M_PI), acos((g / -h))) * 0.3333333333333333));
}

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(fma(2.0, pi, acos(Float64(g / Float64(-h)))) * 0.3333333333333333)))
end

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[(2.0 * Pi + N[ArcCos[N[(g / (-h)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \cos \left(\mathsf{fma}\left(2, \pi, \cos^{-1} \left(\frac{g}{-h}\right)\right) \cdot 0.3333333333333333\right)

Error

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. Simplified1.0

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

  3. Final simplification1.0

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

Reproduce

herbie shell --seed 2022203 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))