2-ancestry mixing, negative discriminant

Average Error: 1.0 → 1.4

Time: 4.2s

Precision: binary64

Cost: 45440

Math

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

↓

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

FPCore

(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
  (*
   (log
    (cbrt
     (exp
      (cos
       (fma 0.3333333333333333 (acos (/ g h)) (* 0.6666666666666666 PI))))))
   3.0)))

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) {
	return 2.0 * (log(cbrt(exp(cos(fma(0.3333333333333333, acos((g / h)), (0.6666666666666666 * ((double) M_PI))))))) * 3.0);
}

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)
	return Float64(2.0 * Float64(log(cbrt(exp(cos(fma(0.3333333333333333, acos(Float64(g / h)), Float64(0.6666666666666666 * pi)))))) * 3.0))
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_] := N[(2.0 * N[(N[Log[N[Power[N[Exp[N[Cos[N[(0.3333333333333333 * N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision] + N[(0.6666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $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))))): 2 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 1.4

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

4. Simplified 1.4

   \[\leadsto 2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right)}}\right) \cdot 3\right)} \]
   Proof
   (*.f64 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 1/3 (acos.f64 (/.f64 g h)) (*.f64 2/3 (PI.f64))))))) 3): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 1/3 (acos.f64 (/.f64 g h)) (Rewrite<= *-commutative_binary64 (*.f64 (PI.f64) 2/3))))))) 3): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/3 (acos.f64 (/.f64 g h))) (*.f64 (PI.f64) 2/3))))))) 3): 1 points increase in error, 1 points decrease in error
   (*.f64 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (acos.f64 (/.f64 g h)) 1/3)) (*.f64 (PI.f64) 2/3)))))) 3): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (PI.f64) 2/3) (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))) 3): 0 points increase in error, 0 points decrease in error
   (*.f64 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))) 3): 0 points increase in error, 1 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 3 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))) (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))) 2))) (log.f64 (cbrt.f64 (exp.f64 (cos.f64 (fma.f64 (PI.f64) 2/3 (*.f64 (acos.f64 (/.f64 g h)) 1/3))))))): 1 points increase in error, 3 points decrease in error

5. Final simplification 1.4

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

Alternatives

Alternative 1
Error	1.4
Cost	39168

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

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

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

Alternative 4
Error	2.3
Cost	19840

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

Reproduce

herbie shell --seed 2022329 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))