Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 9.3

Time: 19.3s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right) \cdot \left(t_0 \cdot -2\right)\right)\\ \mathbf{if}\;J \leq 7.467385102911939 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 3.850756319802687 \cdot 10^{-113}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (hypot 1.0 (/ U (* t_0 (* J 2.0)))) (* t_0 -2.0)))))
   (if (<= J 7.467385102911939e-167)
     t_1
     (if (<= J 3.850756319802687e-113) (- 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 * (hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0));
	double tmp;
	if (J <= 7.467385102911939e-167) {
		tmp = t_1;
	} else if (J <= 3.850756319802687e-113) {
		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 * (Math.hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0));
	double tmp;
	if (J <= 7.467385102911939e-167) {
		tmp = t_1;
	} else if (J <= 3.850756319802687e-113) {
		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 * (math.hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0))
	tmp = 0
	if J <= 7.467385102911939e-167:
		tmp = t_1
	elif J <= 3.850756319802687e-113:
		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(J * Float64(hypot(1.0, Float64(U / Float64(t_0 * Float64(J * 2.0)))) * Float64(t_0 * -2.0)))
	tmp = 0.0
	if (J <= 7.467385102911939e-167)
		tmp = t_1;
	elseif (J <= 3.850756319802687e-113)
		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 * (hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0));
	tmp = 0.0;
	if (J <= 7.467385102911939e-167)
		tmp = t_1;
	elseif (J <= 3.850756319802687e-113)
		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[(J * N[(N[Sqrt[1.0 ^ 2 + N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 7.467385102911939e-167], t$95$1, If[LessEqual[J, 3.850756319802687e-113], (-U), t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if J < 7.4673851029119392e-167 or 3.8507563198026868e-113 < J

   1. Initial program 17.9

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

   2. Simplified 7.9

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      Proof
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (*.f64 J 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (Rewrite<= *-commutative_binary64 (*.f64 2 J))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 42 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 -2 (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J -2) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 J)) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 1 points increase in error, 5 points decrease in error

   if 7.4673851029119392e-167 < J < 3.8507563198026868e-113

   1. Initial program 30.8

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

   2. Taylor expanded in J around 0 41.3

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

   3. Simplified 41.3

      \[\leadsto \color{blue}{-U} \]
      Proof
      (neg.f64 U): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 U)): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 7.467385102911939 \cdot 10^{-167}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;J \leq 3.850756319802687 \cdot 10^{-113}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.5
Cost	14620

\[\begin{array}{l} t_0 := J \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot -2\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{if}\;J \leq -9.687178830394221 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -8.704907557831641 \cdot 10^{-223}:\\ \;\;\;\;\frac{\frac{1 + \cos K}{2}}{\frac{\frac{\frac{U}{J}}{J}}{-2}} - U\\ \mathbf{elif}\;J \leq -8.776417654748253 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.492072057096223 \cdot 10^{-301}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.152650362968972 \cdot 10^{-269}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5.055090786666779 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 3.850756319802687 \cdot 10^{-113}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	26.5
Cost	14292

\[\begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -6.4949539317656895 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 3.485720536501813 \cdot 10^{-69}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.6225029438717393 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.470423735684663 \cdot 10^{-26}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5.236582434870763 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U}{J} \cdot \frac{U}{J}, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	25.9
Cost	7112

\[\begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -6.4949539317656895 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 3.485720536501813 \cdot 10^{-69}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	46.3
Cost	788

\[\begin{array}{l} \mathbf{if}\;U \leq -4.264650270809532 \cdot 10^{+281}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -5.813454331339008 \cdot 10^{+202}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.0194215698806132 \cdot 10^{+67}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.0109298219381062 \cdot 10^{-63}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.8329021101466762 \cdot 10^{+115}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Error	37.5
Cost	588

\[\begin{array}{l} \mathbf{if}\;J \leq -6.351080905809314 \cdot 10^{-34}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -3.777109769446777 \cdot 10^{-68}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.485720536501813 \cdot 10^{-69}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 6
Error	47.2
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022298 
(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)))))