Maksimov and Kolovsky, Equation (3)

Average Error: 17.9 → 8.2

Time: 10.2s

Precision: binary64

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

↓

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq 2.15 \cdot 10^{+232}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, \frac{J}{\frac{U}{J}} \cdot {t_0}^{2}\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))

↓

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= U 2.15e+232)
     (* -2.0 (* (* J t_0) (hypot 1.0 (/ U (* t_0 (* J 2.0))))))
     (* -2.0 (fma 0.5 U (* (/ J (/ U J)) (pow t_0 2.0)))))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + pow((U / ((2.0 * J) * cos((K / 2.0)))), 2.0)));
}

↓

double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (U <= 2.15e+232) {
		tmp = -2.0 * ((J * t_0) * hypot(1.0, (U / (t_0 * (J * 2.0)))));
	} else {
		tmp = -2.0 * fma(0.5, U, ((J / (U / J)) * pow(t_0, 2.0)));
	}
	return tmp;
}

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * cos(Float64(K / 2.0)))) ^ 2.0))))
end

↓

function code(J, K, U)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (U <= 2.15e+232)
		tmp = Float64(-2.0 * Float64(Float64(J * t_0) * hypot(1.0, Float64(U / Float64(t_0 * Float64(J * 2.0))))));
	else
		tmp = Float64(-2.0 * fma(0.5, U, Float64(Float64(J / Float64(U / J)) * (t_0 ^ 2.0))));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, 2.15e+232], N[(-2.0 * N[(N[(J * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(0.5 * U + N[(N[(J / N[(U / J), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 regimes

2. if U < 2.1500000000000001e232

   1. Initial program 16.1

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 6.5

      \[\leadsto \color{blue}{-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]

   3. Applied egg-rr 7.6

      \[\leadsto -2 \cdot \color{blue}{{\left(\sqrt[3]{J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)}\right)\right)}\right)}^{3}} \]

   4. Applied egg-rr 6.5

      \[\leadsto -2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}\right)\right)} \]

   if 2.1500000000000001e232 < U

   1. Initial program 44.4

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 29.6

      \[\leadsto \color{blue}{-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]

   3. Applied egg-rr 30.2

      \[\leadsto -2 \cdot \color{blue}{{\left(\sqrt[3]{J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)}\right)\right)}\right)}^{3}} \]

   4. Applied egg-rr 29.6

      \[\leadsto -2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}\right)\right)} \]

   5. Taylor expanded in J around 0 37.9

      \[\leadsto -2 \cdot \color{blue}{\left(\frac{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U} + 0.5 \cdot U\right)} \]

   6. Simplified 32.7

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(0.5, U, \frac{J}{\frac{U}{J}} \cdot {\cos \left(0.5 \cdot K\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.15 \cdot 10^{+232}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, \frac{J}{\frac{U}{J}} \cdot {\cos \left(K \cdot 0.5\right)}^{2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))