Average Error: 1.0 → 1.0
Time: 6.1s
Precision: binary64
Cost: 26176
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
↓
\[2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \]
(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 PI 0.6666666666666666 (/ (acos (/ (- g) h)) 3.0)))))
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(((double) M_PI), 0.6666666666666666, (acos((-g / h)) / 3.0)));
}

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(fma(pi, 0.6666666666666666, Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
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[(Pi * 0.6666666666666666 + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $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(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\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

    \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \]
    Proof
    (*.f64 2 (cos.f64 (fma.f64 (PI.f64) 2/3 (/.f64 (acos.f64 (/.f64 (neg.f64 g) h)) 3)))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (cos.f64 (fma.f64 (PI.f64) (Rewrite<= metadata-eval (/.f64 2 3)) (/.f64 (acos.f64 (/.f64 (neg.f64 g) h)) 3)))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (cos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (PI.f64) (/.f64 2 3)) (/.f64 (acos.f64 (/.f64 (neg.f64 g) h)) 3))))): 3 points increase in error, 0 points decrease in error
    (*.f64 2 (cos.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 3) (PI.f64))) (/.f64 (acos.f64 (/.f64 (neg.f64 g) h)) 3)))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (cos.f64 (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 3 (PI.f64)))) (/.f64 (acos.f64 (/.f64 (neg.f64 g) h)) 3)))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (cos.f64 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 (PI.f64)) 3)) (/.f64 (acos.f64 (/.f64 (neg.f64 g) h)) 3)))): 0 points increase in error, 0 points decrease in error

  3. Final simplification1.0

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

Alternatives

Alternative 1
Error1.0
Cost19904
\[2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3} + \pi \cdot 0.6666666666666666\right) \]
Alternative 2
Error2.4
Cost19840
\[2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right) \]

Error

Reproduce

herbie shell --seed 2022293 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))