2-ancestry mixing, negative discriminant

Average Error: 1.0 → 1.3

Time: 6.7s

Precision: binary64

Cost: 97024

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}}\\ 2 \cdot \left(\log \left({t_0}^{2}\right) + \log t_0\right) \end{array} \]

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
         (cbrt
          (exp
           (cos
            (fma
             PI
             0.6666666666666666
             (* (acos (/ g h)) 0.3333333333333333)))))))
   (* 2.0 (+ (log (pow t_0 2.0)) (log 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 = cbrt(exp(cos(fma(((double) M_PI), 0.6666666666666666, (acos((g / h)) * 0.3333333333333333)))));
	return 2.0 * (log(pow(t_0, 2.0)) + log(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 = cbrt(exp(cos(fma(pi, 0.6666666666666666, Float64(acos(Float64(g / h)) * 0.3333333333333333)))))
	return Float64(2.0 * Float64(log((t_0 ^ 2.0)) + log(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[Power[N[Exp[N[Cos[N[(Pi * 0.6666666666666666 + N[(N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(2.0 * N[(N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $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

   \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \]
   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))))): 4 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. Applied egg-rr 1.3

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

4. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.2
Cost	71424

\[\begin{array}{l} t_0 := \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)\\ 2 \cdot \left(\sqrt[3]{t_0} \cdot \sqrt[3]{{t_0}^{2}}\right) \end{array} \]

Alternative 2
Error	1.2
Cost	58880

\[\begin{array}{l} t_0 := \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333 + \pi \cdot 0.6666666666666666\right)\\ 2 \cdot \left(\sqrt[3]{t_0} \cdot \sqrt[3]{{t_0}^{2}}\right) \end{array} \]

Alternative 3
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 4
Error	1.0
Cost	19904

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

Alternative 5
Error	2.2
Cost	19840

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

Reproduce

herbie shell --seed 2022283 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))