Maksimov and Kolovsky, Equation (3)

Average Accuracy: 72.0% → 86.6%

Time: 19.4s

Precision: binary64

Cost: 20748

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J \cdot \left(-2 \cdot t_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\\ \mathbf{if}\;J \leq -4.6 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 9.8 \cdot 10^{-307}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{-220}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 2.0)))
        (t_1 (* (* J (* -2.0 t_0)) (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))
   (if (<= J -4.6e-262)
     t_1
     (if (<= J 9.8e-307) U (if (<= J 2.7e-220) (- U) t_1)))))

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 / 2.0));
	double t_1 = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	double tmp;
	if (J <= -4.6e-262) {
		tmp = t_1;
	} else if (J <= 9.8e-307) {
		tmp = U;
	} else if (J <= 2.7e-220) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (J * (-2.0 * t_0)) * Math.hypot(1.0, (U / (J * (2.0 * t_0))));
	double tmp;
	if (J <= -4.6e-262) {
		tmp = t_1;
	} else if (J <= 9.8e-307) {
		tmp = U;
	} else if (J <= 2.7e-220) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * math.cos((K / 2.0)))), 2.0)))

↓

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = (J * (-2.0 * t_0)) * math.hypot(1.0, (U / (J * (2.0 * t_0))))
	tmp = 0
	if J <= -4.6e-262:
		tmp = t_1
	elif J <= 9.8e-307:
		tmp = U
	elif J <= 2.7e-220:
		tmp = -U
	else:
		tmp = t_1
	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 / 2.0))
	t_1 = Float64(Float64(J * Float64(-2.0 * t_0)) * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))
	tmp = 0.0
	if (J <= -4.6e-262)
		tmp = t_1;
	elseif (J <= 9.8e-307)
		tmp = U;
	elseif (J <= 2.7e-220)
		tmp = Float64(-U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((U / ((2.0 * J) * cos((K / 2.0)))) ^ 2.0)));
end

↓

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	tmp = 0.0;
	if (J <= -4.6e-262)
		tmp = t_1;
	elseif (J <= 9.8e-307)
		tmp = U;
	elseif (J <= 2.7e-220)
		tmp = -U;
	else
		tmp = t_1;
	end
	tmp_2 = 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 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.6e-262], t$95$1, If[LessEqual[J, 9.8e-307], U, If[LessEqual[J, 2.7e-220], (-U), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.6000000000000002e-262 or 2.7e-220 < J

   1. Initial program 76.5%

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

   2. Simplified 92.4%

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

   if -4.6000000000000002e-262 < J < 9.8000000000000005e-307

   1. Initial program 27.3%

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

   2. Simplified 36.2%

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

   3. Taylor expanded in U around -inf 82.3%

      \[\leadsto \color{blue}{U} \]

   if 9.8000000000000005e-307 < J < 2.7e-220

   1. Initial program 33.3%

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

   2. Simplified 44.4%

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

   3. Taylor expanded in J around 0 67.8%

      \[\leadsto \color{blue}{-1 \cdot U} \]

   4. Simplified 67.8%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.6 \cdot 10^{-262}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{elif}\;J \leq 9.8 \cdot 10^{-307}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{-220}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.5%
Cost	20748

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(t_0 \cdot \left(-2 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right)\\ \mathbf{if}\;J \leq -7.8 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 1.08 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.5 \cdot 10^{-219}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	66.9%
Cost	14356

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ t_1 := \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot t_0\\ \mathbf{if}\;J \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 1.08 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6 \cdot 10^{-220}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.4 \cdot 10^{-187}:\\ \;\;\;\;t_0 \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{-68}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	59.6%
Cost	8360

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right) \cdot \left(J \cdot -2\right)\\ t_1 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.42 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -102000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -0.00125:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -3.8 \cdot 10^{-152}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6.8 \cdot 10^{-206}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{-217}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5.5 \cdot 10^{-187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{-88}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	55.1%
Cost	7508

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.6 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4.4 \cdot 10^{-151}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -7.8 \cdot 10^{-206}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	38.1%
Cost	1368

\[\begin{array}{l} \mathbf{if}\;J \leq -9 \cdot 10^{+74}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -3.6 \cdot 10^{-152}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.7 \cdot 10^{-206}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 0.000465:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5.5 \cdot 10^{+34}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq 3.8 \cdot 10^{+72}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{+94}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 6
Accuracy	38.3%
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -1.25 \cdot 10^{+75}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -3.7 \cdot 10^{-152}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.45 \cdot 10^{-206}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 9 \cdot 10^{+72}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3 \cdot 10^{+91}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 7
Accuracy	26.4%
Cost	788

\[\begin{array}{l} \mathbf{if}\;J \leq -5.6 \cdot 10^{-147}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -7.8 \cdot 10^{-206}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9.8 \cdot 10^{-307}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{+72}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.5 \cdot 10^{+135}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Accuracy	26.8%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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)))))