Average Error: 1.0 → 1.3
Time: 6.8s
Precision: binary64
Cost: 169920
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
↓
\[\begin{array}{l} t_0 := \cos \left(0.6666666666666666 \cdot \pi\right)\\ t_1 := \cos^{-1} \left(\frac{g}{h}\right)\\ t_2 := 0.3333333333333333 \cdot t_1\\ t_3 := \sin t_2 \cdot \sin \left(0.6666666666666666 \cdot \pi\right)\\ t_4 := \cos t_2\\ 2 \cdot \frac{{\left(t_0 \cdot t_4\right)}^{3} - {t_3}^{3}}{\mathsf{fma}\left(t_3, \cos \left(0.6666666666666666 \cdot \pi + t_1 \cdot -0.3333333333333333\right), t_0 \cdot \left(t_0 \cdot \left(t_4 \cdot t_4\right)\right)\right)} \end{array} \]
(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)))
        (t_1 (acos (/ g h)))
        (t_2 (* 0.3333333333333333 t_1))
        (t_3 (* (sin t_2) (sin (* 0.6666666666666666 PI))))
        (t_4 (cos t_2)))
   (*
    2.0
    (/
     (- (pow (* t_0 t_4) 3.0) (pow t_3 3.0))
     (fma
      t_3
      (cos (+ (* 0.6666666666666666 PI) (* t_1 -0.3333333333333333)))
      (* t_0 (* t_0 (* t_4 t_4))))))))
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)));
	double t_1 = acos((g / h));
	double t_2 = 0.3333333333333333 * t_1;
	double t_3 = sin(t_2) * sin((0.6666666666666666 * ((double) M_PI)));
	double t_4 = cos(t_2);
	return 2.0 * ((pow((t_0 * t_4), 3.0) - pow(t_3, 3.0)) / fma(t_3, cos(((0.6666666666666666 * ((double) M_PI)) + (t_1 * -0.3333333333333333))), (t_0 * (t_0 * (t_4 * t_4)))));
}

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(0.6666666666666666 * pi))
	t_1 = acos(Float64(g / h))
	t_2 = Float64(0.3333333333333333 * t_1)
	t_3 = Float64(sin(t_2) * sin(Float64(0.6666666666666666 * pi)))
	t_4 = cos(t_2)
	return Float64(2.0 * Float64(Float64((Float64(t_0 * t_4) ^ 3.0) - (t_3 ^ 3.0)) / fma(t_3, cos(Float64(Float64(0.6666666666666666 * pi) + Float64(t_1 * -0.3333333333333333))), Float64(t_0 * Float64(t_0 * Float64(t_4 * t_4))))))
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_] := Block[{t$95$0 = N[Cos[N[(0.6666666666666666 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[N[(0.6666666666666666 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$2], $MachinePrecision]}, N[(2.0 * N[(N[(N[Power[N[(t$95$0 * t$95$4), $MachinePrecision], 3.0], $MachinePrecision] - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[Cos[N[(N[(0.6666666666666666 * Pi), $MachinePrecision] + N[(t$95$1 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

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\right)\\
t_1 := \cos^{-1} \left(\frac{g}{h}\right)\\
t_2 := 0.3333333333333333 \cdot t_1\\
t_3 := \sin t_2 \cdot \sin \left(0.6666666666666666 \cdot \pi\right)\\
t_4 := \cos t_2\\
2 \cdot \frac{{\left(t_0 \cdot t_4\right)}^{3} - {t_3}^{3}}{\mathsf{fma}\left(t_3, \cos \left(0.6666666666666666 \cdot \pi + t_1 \cdot -0.3333333333333333\right), t_0 \cdot \left(t_0 \cdot \left(t_4 \cdot t_4\right)\right)\right)}
\end{array}

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

    \[\leadsto \color{blue}{2 \cdot \cos \left(0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]
    Proof
    (*.f64 2 (cos.f64 (+.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 (*.f64 (Rewrite<= metadata-eval (/.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, 4 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)))): 4 points increase in error, 0 points decrease in error

  3. Applied egg-rr2.8

    \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - {\left(\sin \left(0.6666666666666666 \cdot \pi\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}^{3}}{\left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right) \cdot \left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right) + \left(\left(\sin \left(0.6666666666666666 \cdot \pi\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(0.6666666666666666 \cdot \pi\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right) + \left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(0.6666666666666666 \cdot \pi\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)\right)}} \]

  4. Simplified2.2

    \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right)}^{3} - {\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right)\right)}^{3}}{\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right), \mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right), \sin \left(0.6666666666666666 \cdot \pi\right), \cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right), \cos \left(0.6666666666666666 \cdot \pi\right) \cdot \left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \left(\cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right)\right)}} \]
    Proof
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h))))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64)))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64)))) (fma.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64)))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64)))) (fma.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (sin.f64 (*.f64 2/3 (PI.f64)))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64)))) (fma.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 19 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64)))) (fma.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 19 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (sin.f64 (*.f64 2/3 (PI.f64)))) (fma.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (fma.f64 (sin.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (fma.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 19 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (fma.f64 (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)) (sin.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 19 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)) (sin.f64 (*.f64 2/3 (PI.f64)))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (+.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (cos.f64 (*.f64 1/3 (acos.f64 (/.f64 g h)))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (+.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (*.f64 (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (+.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 2/3 (PI.f64)))) (*.f64 (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (fma.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (+.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (+.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))) (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))))): 19 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (+.f64 (*.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))) (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))): 0 points increase in error, 19 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))) (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (/.f64 (-.f64 (pow.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3) (pow.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) 3)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (+.f64 (*.f64 (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))) (*.f64 (*.f64 (cos.f64 (*.f64 2/3 (PI.f64))) (cos.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))) (*.f64 (sin.f64 (*.f64 2/3 (PI.f64))) (sin.f64 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr1.3

    \[\leadsto 2 \cdot \frac{{\left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right)}^{3} - {\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right)\right)}^{3}}{\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right), \color{blue}{\cos \left(0.6666666666666666 \cdot \pi + -0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)}, \cos \left(0.6666666666666666 \cdot \pi\right) \cdot \left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \left(\cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right)\right)} \]

  6. Final simplification1.3

    \[\leadsto 2 \cdot \frac{{\left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right)}^{3} - {\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right)\right)}^{3}}{\mathsf{fma}\left(\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \sin \left(0.6666666666666666 \cdot \pi\right), \cos \left(0.6666666666666666 \cdot \pi + \cos^{-1} \left(\frac{g}{h}\right) \cdot -0.3333333333333333\right), \cos \left(0.6666666666666666 \cdot \pi\right) \cdot \left(\cos \left(0.6666666666666666 \cdot \pi\right) \cdot \left(\cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \cdot \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right)\right)} \]

Alternatives

Alternative 1
Error1.0
Cost26176
\[2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \]
Alternative 2
Error1.0
Cost19904
\[2 \cdot \cos \left(0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
Alternative 3
Error2.2
Cost19840
\[2 \cdot \cos \left(0.6666666666666666 \cdot \pi + 0.3333333333333333 \cdot \cos^{-1} \left(\frac{g}{h}\right)\right) \]

Error

Reproduce

herbie shell --seed 2022341 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))