Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.1%

Time: 12.5s

Alternatives: 10

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

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

Alternative 1: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_3 := t\_2 \cdot \sqrt{1 + {t\_1}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_2 \cdot \mathsf{hypot}\left(1, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (/ U_m (* t_0 (* J_m 2.0))))
        (t_2 (* (* -2.0 J_m) t_0))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow t_1 2.0))))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_3 2e+294) (* t_2 (hypot 1.0 t_1)) U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = U_m / (t_0 * (J_m * 2.0));
	double t_2 = (-2.0 * J_m) * t_0;
	double t_3 = t_2 * sqrt((1.0 + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_3 <= 2e+294) {
		tmp = t_2 * hypot(1.0, t_1);
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U_m / (t_0 * (J_m * 2.0));
	double t_2 = (-2.0 * J_m) * t_0;
	double t_3 = t_2 * Math.sqrt((1.0 + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - U_m;
	} else if (t_3 <= 2e+294) {
		tmp = t_2 * Math.hypot(1.0, t_1);
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = U_m / (t_0 * (J_m * 2.0))
	t_2 = (-2.0 * J_m) * t_0
	t_3 = t_2 * math.sqrt((1.0 + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = 0.0 - U_m
	elif t_3 <= 2e+294:
		tmp = t_2 * math.hypot(1.0, t_1)
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U_m / Float64(t_0 * Float64(J_m * 2.0)))
	t_2 = Float64(Float64(-2.0 * J_m) * t_0)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_3 <= 2e+294)
		tmp = Float64(t_2 * hypot(1.0, t_1));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = U_m / (t_0 * (J_m * 2.0));
	t_2 = (-2.0 * J_m) * t_0;
	t_3 = t_2 * sqrt((1.0 + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = 0.0 - U_m;
	elseif (t_3 <= 2e+294)
		tmp = t_2 * hypot(1.0, t_1);
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2e+294], N[(t$95$2 * N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision], U$95$m]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 63.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 63.5%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 63.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000013e294

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. associate-*l* N/A

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

   4. Applied egg-rr 99.8%

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

   if 2.00000000000000013e294 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 11.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 32.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.39:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.52:\\ \;\;\;\;t\_0 \cdot \left(-2 \cdot J\_m + -0.25 \cdot \frac{U\_m \cdot U\_m}{J\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= t_0 -0.39)
      U_m
      (if (<= t_0 0.52)
        (* t_0 (+ (* -2.0 J_m) (* -0.25 (/ (* U_m U_m) J_m))))
        (if (<= t_0 0.9995)
          (- 0.0 U_m)
          (* (* -2.0 J_m) (hypot 1.0 (* 0.5 (/ U_m J_m))))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.39) {
		tmp = U_m;
	} else if (t_0 <= 0.52) {
		tmp = t_0 * ((-2.0 * J_m) + (-0.25 * ((U_m * U_m) / J_m)));
	} else if (t_0 <= 0.9995) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 * (U_m / J_m)));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.39) {
		tmp = U_m;
	} else if (t_0 <= 0.52) {
		tmp = t_0 * ((-2.0 * J_m) + (-0.25 * ((U_m * U_m) / J_m)));
	} else if (t_0 <= 0.9995) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, (0.5 * (U_m / J_m)));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.39:
		tmp = U_m
	elif t_0 <= 0.52:
		tmp = t_0 * ((-2.0 * J_m) + (-0.25 * ((U_m * U_m) / J_m)))
	elif t_0 <= 0.9995:
		tmp = 0.0 - U_m
	else:
		tmp = (-2.0 * J_m) * math.hypot(1.0, (0.5 * (U_m / J_m)))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.39)
		tmp = U_m;
	elseif (t_0 <= 0.52)
		tmp = Float64(t_0 * Float64(Float64(-2.0 * J_m) + Float64(-0.25 * Float64(Float64(U_m * U_m) / J_m))));
	elseif (t_0 <= 0.9995)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(0.5 * Float64(U_m / J_m))));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.39)
		tmp = U_m;
	elseif (t_0 <= 0.52)
		tmp = t_0 * ((-2.0 * J_m) + (-0.25 * ((U_m * U_m) / J_m)));
	elseif (t_0 <= 0.9995)
		tmp = 0.0 - U_m;
	else
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 * (U_m / J_m)));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$0, -0.39], U$95$m, If[LessEqual[t$95$0, 0.52], N[(t$95$0 * N[(N[(-2.0 * J$95$m), $MachinePrecision] + N[(-0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(0.0 - U$95$m), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.39000000000000001

   1. Initial program 61.5%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 30.3%

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

   if -0.39000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.52000000000000002

   1. Initial program 81.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. metadata-eval N/A

         \[\leadsto \sqrt{1 \cdot 1 + \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)} \cdot \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}} \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot J\right) \cdot \cos \left(\frac{\color{blue}{K}}{2}\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{1 \cdot 1 + \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)} \cdot \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}} \cdot \left(\left(\mathsf{neg}\left(2 \cdot J\right)\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{1 \cdot 1 + \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)} \cdot \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}} \cdot \left(\left(\mathsf{neg}\left(J \cdot 2\right)\right) \cdot \cos \left(\frac{\color{blue}{K}}{2}\right)\right) \]

   4. Applied egg-rr 86.5%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}\right), \mathsf{*.f64}\left(J, -2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{hypot.f64}\left(1, \left(\frac{\frac{1}{2} \cdot U}{J}\right)\right), \mathsf{*.f64}\left(J, -2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot U\right), J\right)\right), \mathsf{*.f64}\left(J, -2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]

      3. *-lowering-*.f64 66.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, U\right), J\right)\right), \mathsf{*.f64}\left(J, -2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]

   7. Simplified 66.6%

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

   8. Taylor expanded in U around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J + \frac{-1}{4} \cdot \frac{{U}^{2}}{J}\right)}, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot J\right), \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{J}\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(K, 2\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{J}\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\color{blue}{K}, 2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(\frac{{U}^{2}}{J}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, \color{blue}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\left({U}^{2}\right), J\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\left(U \cdot U\right), J\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]

      6. *-lowering-*.f64 64.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, U\right), J\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right) \]

   10. Simplified 64.5%

       \[\leadsto \color{blue}{\left(-2 \cdot J + -0.25 \cdot \frac{U \cdot U}{J}\right)} \cdot \cos \left(\frac{K}{2}\right) \]

   if 0.52000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99950000000000006

   1. Initial program 62.5%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 36.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 36.3%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 36.3%

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

   if 0.99950000000000006 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 74.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 91.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 91.3%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{U}{J}\right)}\right)\right)\right) \]

      2. /-lowering-/.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, \color{blue}{J}\right)\right)\right)\right) \]

   7. Simplified 90.0%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, J\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 90.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{hypot.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, J\right)\right)\right)\right) \]

   10. Simplified 90.7%

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.39:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.52:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J + -0.25 \cdot \frac{U \cdot U}{J}\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.9995:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.39:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.205:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= t_0 -0.39)
      U_m
      (if (<= t_0 0.205)
        (* (* -2.0 J_m) (cos (* K 0.5)))
        (if (<= t_0 0.9995)
          (- 0.0 U_m)
          (* (* -2.0 J_m) (hypot 1.0 (* 0.5 (/ U_m J_m))))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.39) {
		tmp = U_m;
	} else if (t_0 <= 0.205) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else if (t_0 <= 0.9995) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 * (U_m / J_m)));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.39) {
		tmp = U_m;
	} else if (t_0 <= 0.205) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else if (t_0 <= 0.9995) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, (0.5 * (U_m / J_m)));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.39:
		tmp = U_m
	elif t_0 <= 0.205:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	elif t_0 <= 0.9995:
		tmp = 0.0 - U_m
	else:
		tmp = (-2.0 * J_m) * math.hypot(1.0, (0.5 * (U_m / J_m)))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.39)
		tmp = U_m;
	elseif (t_0 <= 0.205)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	elseif (t_0 <= 0.9995)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(0.5 * Float64(U_m / J_m))));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.39)
		tmp = U_m;
	elseif (t_0 <= 0.205)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	elseif (t_0 <= 0.9995)
		tmp = 0.0 - U_m;
	else
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 * (U_m / J_m)));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$0, -0.39], U$95$m, If[LessEqual[t$95$0, 0.205], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(0.0 - U$95$m), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.39000000000000001

   1. Initial program 61.5%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 30.3%

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

   if -0.39000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.204999999999999988

   1. Initial program 87.7%

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

   3. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(J \cdot \color{blue}{-2}\right)\right) \]

      7. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(J, \color{blue}{-2}\right)\right) \]

   5. Simplified 70.3%

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

   if 0.204999999999999988 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99950000000000006

   1. Initial program 63.8%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 37.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 37.8%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 37.8%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 37.8%

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

   if 0.99950000000000006 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 74.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 91.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 91.3%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{U}{J}\right)}\right)\right)\right) \]

      2. /-lowering-/.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, \color{blue}{J}\right)\right)\right)\right) \]

   7. Simplified 90.0%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, J\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 90.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{hypot.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, J\right)\right)\right)\right) \]

   10. Simplified 90.7%

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.39:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.205:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.9995:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.25 \cdot 10^{+120}:\\ \;\;\;\;t\_0 \cdot \left(\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= U_m 1.25e+120)
      (* t_0 (* (* -2.0 J_m) (hypot 1.0 (/ U_m (* t_0 (* J_m 2.0))))))
      (- 0.0 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 1.25e+120) {
		tmp = t_0 * ((-2.0 * J_m) * hypot(1.0, (U_m / (t_0 * (J_m * 2.0)))));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U_m <= 1.25e+120) {
		tmp = t_0 * ((-2.0 * J_m) * Math.hypot(1.0, (U_m / (t_0 * (J_m * 2.0)))));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 1.25e+120:
		tmp = t_0 * ((-2.0 * J_m) * math.hypot(1.0, (U_m / (t_0 * (J_m * 2.0)))))
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 1.25e+120)
		tmp = Float64(t_0 * Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))))));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 1.25e+120)
		tmp = t_0 * ((-2.0 * J_m) * hypot(1.0, (U_m / (t_0 * (J_m * 2.0)))));
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 1.25e+120], N[(t$95$0 * N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.25000000000000005e120

   1. Initial program 75.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. metadata-eval N/A

         \[\leadsto \sqrt{1 \cdot 1 + \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)} \cdot \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}} \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot J\right) \cdot \cos \left(\frac{\color{blue}{K}}{2}\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{1 \cdot 1 + \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)} \cdot \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}} \cdot \left(\left(\mathsf{neg}\left(2 \cdot J\right)\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{1 \cdot 1 + \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)} \cdot \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}} \cdot \left(\left(\mathsf{neg}\left(J \cdot 2\right)\right) \cdot \cos \left(\frac{\color{blue}{K}}{2}\right)\right) \]

   4. Applied egg-rr 90.8%

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

   if 1.25000000000000005e120 < U

   1. Initial program 39.3%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 59.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 59.2%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.25 \cdot 10^{+120}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 8.5 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= U_m 8.5e+119)
      (* (* (* -2.0 J_m) t_0) (hypot 1.0 (/ (/ U_m (* J_m 2.0)) t_0)))
      (- 0.0 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 8.5e+119) {
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U_m <= 8.5e+119) {
		tmp = ((-2.0 * J_m) * t_0) * Math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 8.5e+119:
		tmp = ((-2.0 * J_m) * t_0) * math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0))
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 8.5e+119)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * hypot(1.0, Float64(Float64(U_m / Float64(J_m * 2.0)) / t_0)));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 8.5e+119)
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 8.5e+119], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 8.49999999999999997e119

   1. Initial program 75.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 90.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 90.8%

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

   if 8.49999999999999997e119 < U

   1. Initial program 39.3%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 59.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 59.2%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 8.5 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 3e+104)
    (* (* (* -2.0 J_m) (cos (/ K 2.0))) (hypot 1.0 (* 0.5 (/ U_m J_m))))
    (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 3e+104) {
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * hypot(1.0, (0.5 * (U_m / J_m)));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 3e+104) {
		tmp = ((-2.0 * J_m) * Math.cos((K / 2.0))) * Math.hypot(1.0, (0.5 * (U_m / J_m)));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 3e+104:
		tmp = ((-2.0 * J_m) * math.cos((K / 2.0))) * math.hypot(1.0, (0.5 * (U_m / J_m)))
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 3e+104)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K / 2.0))) * hypot(1.0, Float64(0.5 * Float64(U_m / J_m))));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 3e+104)
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * hypot(1.0, (0.5 * (U_m / J_m)));
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 3e+104], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 2.99999999999999969e104

   1. Initial program 76.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 90.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 90.7%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{U}{J}\right)}\right)\right)\right) \]

      2. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, \color{blue}{J}\right)\right)\right)\right) \]

   7. Simplified 74.3%

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

   if 2.99999999999999969e104 < U

   1. Initial program 38.3%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 60.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 60.3%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 60.3%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1.35e+104) (* (* -2.0 J_m) (cos (* K 0.5))) (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.35e+104) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.35d+104) then
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.35e+104) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.35e+104:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.35e+104)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.35e+104)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.35e+104], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.34999999999999992e104

   1. Initial program 76.1%

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

   3. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(J \cdot \color{blue}{-2}\right)\right) \]

      7. *-lowering-*.f64 56.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(J, \color{blue}{-2}\right)\right) \]

   5. Simplified 56.1%

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

   if 1.34999999999999992e104 < U

   1. Initial program 38.3%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 60.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 60.3%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 60.3%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.7% accurate, 32.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 5.8 \cdot 10^{+25}:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;K \leq 1.75 \cdot 10^{+99}:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= K 5.8e+25) (- 0.0 U_m) (if (<= K 1.75e+99) U_m (- 0.0 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 5.8e+25) {
		tmp = 0.0 - U_m;
	} else if (K <= 1.75e+99) {
		tmp = U_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (k <= 5.8d+25) then
        tmp = 0.0d0 - u_m
    else if (k <= 1.75d+99) then
        tmp = u_m
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 5.8e+25) {
		tmp = 0.0 - U_m;
	} else if (K <= 1.75e+99) {
		tmp = U_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if K <= 5.8e+25:
		tmp = 0.0 - U_m
	elif K <= 1.75e+99:
		tmp = U_m
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (K <= 5.8e+25)
		tmp = Float64(0.0 - U_m);
	elseif (K <= 1.75e+99)
		tmp = U_m;
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (K <= 5.8e+25)
		tmp = 0.0 - U_m;
	elseif (K <= 1.75e+99)
		tmp = U_m;
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[K, 5.8e+25], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[K, 1.75e+99], U$95$m, N[(0.0 - U$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 5.7999999999999998e25 or 1.7499999999999999e99 < K

   1. Initial program 70.7%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 31.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 31.9%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 31.9%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 31.9%

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

   if 5.7999999999999998e25 < K < 1.7499999999999999e99

   1. Initial program 74.5%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 22.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 5.8 \cdot 10^{+25}:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;K \leq 1.75 \cdot 10^{+99}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.6% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.02 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1.02e-49) (* -2.0 J_m) (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.02e-49) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.02d-49) then
        tmp = (-2.0d0) * j_m
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.02e-49) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.02e-49:
		tmp = -2.0 * J_m
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.02e-49)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.02e-49)
		tmp = -2.0 * J_m;
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.02e-49], N[(-2.0 * J$95$m), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.02000000000000009e-49

   1. Initial program 78.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 90.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 90.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{U}{J}\right)}\right)\right)\right) \]

      2. /-lowering-/.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, \color{blue}{J}\right)\right)\right)\right) \]

   7. Simplified 74.4%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, J\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{hypot.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(U, J\right)\right)\right)\right) \]

   10. Simplified 55.8%

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

   11. Taylor expanded in J around inf

       \[\leadsto \color{blue}{-2 \cdot J} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 35.1%

          \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{J}\right) \]

   13. Simplified 35.1%

       \[\leadsto \color{blue}{-2 \cdot J} \]

   if 1.02000000000000009e-49 < U

   1. Initial program 52.4%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 52.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 52.6%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.02 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 13.8% accurate, 420.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s U_m))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * u_m
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * U_m

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * U_m)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * U$95$m), $MachinePrecision]

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 22.6%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))))