Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.6%

Time: 13.3s

Alternatives: 11

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

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

Java

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

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

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

Java

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

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+14} \lor \neg \left(t_1 \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -2e+14) (not (<= t_1 5e-16)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -2e+14) || !(t_1 <= 5e-16)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-2d+14)) .or. (.not. (t_1 <= 5d-16))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -2e+14) || !(t_1 <= 5e-16)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -2e+14) or not (t_1 <= 5e-16):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -2e+14) || !(t_1 <= 5e-16))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -2e+14) || ~((t_1 <= 5e-16)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+14], N[Not[LessEqual[t$95$1, 5e-16]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -2e14 or 5.0000000000000004e-16 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -2e14 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000004e-16

   1. Initial program 73.2%

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

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -2 \cdot 10^{+14} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\\ \mathbf{if}\;t_0 \leq -0.915:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.49:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.01:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (+
          U
          (*
           (* J (pow l 3.0))
           (+ 0.3333333333333333 (* -0.041666666666666664 (* K K)))))))
   (if (<= t_0 -0.915)
     t_1
     (if (<= t_0 -0.49)
       (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
       (if (<= t_0 -0.01) t_1 (+ U (* (sinh l) (* J 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + ((J * pow(l, 3.0)) * (0.3333333333333333 + (-0.041666666666666664 * (K * K))));
	double tmp;
	if (t_0 <= -0.915) {
		tmp = t_1;
	} else if (t_0 <= -0.49) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else {
		tmp = U + (sinh(l) * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + ((j * (l ** 3.0d0)) * (0.3333333333333333d0 + ((-0.041666666666666664d0) * (k * k))))
    if (t_0 <= (-0.915d0)) then
        tmp = t_1
    else if (t_0 <= (-0.49d0)) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else if (t_0 <= (-0.01d0)) then
        tmp = t_1
    else
        tmp = u + (sinh(l) * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + ((J * Math.pow(l, 3.0)) * (0.3333333333333333 + (-0.041666666666666664 * (K * K))));
	double tmp;
	if (t_0 <= -0.915) {
		tmp = t_1;
	} else if (t_0 <= -0.49) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else {
		tmp = U + (Math.sinh(l) * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + ((J * math.pow(l, 3.0)) * (0.3333333333333333 + (-0.041666666666666664 * (K * K))))
	tmp = 0
	if t_0 <= -0.915:
		tmp = t_1
	elif t_0 <= -0.49:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	elif t_0 <= -0.01:
		tmp = t_1
	else:
		tmp = U + (math.sinh(l) * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(Float64(J * (l ^ 3.0)) * Float64(0.3333333333333333 + Float64(-0.041666666666666664 * Float64(K * K)))))
	tmp = 0.0
	if (t_0 <= -0.915)
		tmp = t_1;
	elseif (t_0 <= -0.49)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	elseif (t_0 <= -0.01)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(sinh(l) * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + ((J * (l ^ 3.0)) * (0.3333333333333333 + (-0.041666666666666664 * (K * K))));
	tmp = 0.0;
	if (t_0 <= -0.915)
		tmp = t_1;
	elseif (t_0 <= -0.49)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	elseif (t_0 <= -0.01)
		tmp = t_1;
	else
		tmp = U + (sinh(l) * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 + N[(-0.041666666666666664 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.915], t$95$1, If[LessEqual[t$95$0, -0.49], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], t$95$1, N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.91500000000000004 or -0.48999999999999999 < (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 92.1%

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

   2. Taylor expanded in l around 0 91.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 83.3%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 83.3%

         \[\leadsto \color{blue}{\left(\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 83.3%

         \[\leadsto \left(\color{blue}{\left({\ell}^{3} \cdot 0.3333333333333333\right)} \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 83.3%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(0.3333333333333333 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(0.3333333333333333 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 12.4%

      \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot \left({K}^{2} \cdot \left({\ell}^{3} \cdot J\right)\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   7. Step-by-step derivation

      1. +-commutative 12.4%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + -0.041666666666666664 \cdot \left({K}^{2} \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]

      2. unpow2 12.4%

         \[\leadsto \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + -0.041666666666666664 \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 12.4%

         \[\leadsto \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + \color{blue}{\left(-0.041666666666666664 \cdot \left(K \cdot K\right)\right) \cdot \left({\ell}^{3} \cdot J\right)}\right) + U \]

      4. distribute-rgt-out 83.0%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot J\right) \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)} + U \]

   8. Simplified 83.0%

      \[\leadsto \color{blue}{\left({\ell}^{3} \cdot J\right) \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)} + U \]

   if -0.91500000000000004 < (cos.f64 (/.f64 K 2)) < -0.48999999999999999

   1. Initial program 86.9%

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

   2. Taylor expanded in l around 0 95.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 82.2%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 82.2%

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

      2. *-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-*l* 82.2%

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

      5. *-commutative 82.2%

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

   5. Simplified 82.2%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0 86.5%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
   3. Step-by-step derivation

      1. expm1-log1p-u 64.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 63.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. sinh-undef 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot J\right)} - 1\right) + U \]

   4. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 69.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)\right)} + U \]

      2. expm1-log1p 96.8%

         \[\leadsto \color{blue}{\left(2 \cdot \sinh \ell\right) \cdot J} + U \]

      3. *-commutative 96.8%

         \[\leadsto \color{blue}{\left(\sinh \ell \cdot 2\right)} \cdot J + U \]

      4. associate-*l* 96.8%

         \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]

   6. Simplified 96.8%

      \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.915:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.49:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 3: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \mathbf{if}\;t_0 \leq -0.915:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.65:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.01:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (+ U (* J (* -0.25 (* K (* l K)))))))
   (if (<= t_0 -0.915)
     t_1
     (if (<= t_0 -0.65)
       (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
       (if (<= t_0 -0.01) t_1 (+ U (* (sinh l) (* J 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (J * (-0.25 * (K * (l * K))));
	double tmp;
	if (t_0 <= -0.915) {
		tmp = t_1;
	} else if (t_0 <= -0.65) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else {
		tmp = U + (sinh(l) * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (j * ((-0.25d0) * (k * (l * k))))
    if (t_0 <= (-0.915d0)) then
        tmp = t_1
    else if (t_0 <= (-0.65d0)) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else if (t_0 <= (-0.01d0)) then
        tmp = t_1
    else
        tmp = u + (sinh(l) * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (J * (-0.25 * (K * (l * K))));
	double tmp;
	if (t_0 <= -0.915) {
		tmp = t_1;
	} else if (t_0 <= -0.65) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else {
		tmp = U + (Math.sinh(l) * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (J * (-0.25 * (K * (l * K))))
	tmp = 0
	if t_0 <= -0.915:
		tmp = t_1
	elif t_0 <= -0.65:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	elif t_0 <= -0.01:
		tmp = t_1
	else:
		tmp = U + (math.sinh(l) * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(J * Float64(-0.25 * Float64(K * Float64(l * K)))))
	tmp = 0.0
	if (t_0 <= -0.915)
		tmp = t_1;
	elseif (t_0 <= -0.65)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	elseif (t_0 <= -0.01)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(sinh(l) * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (J * (-0.25 * (K * (l * K))));
	tmp = 0.0;
	if (t_0 <= -0.915)
		tmp = t_1;
	elseif (t_0 <= -0.65)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	elseif (t_0 <= -0.01)
		tmp = t_1;
	else
		tmp = U + (sinh(l) * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(-0.25 * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.915], t$95$1, If[LessEqual[t$95$0, -0.65], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], t$95$1, N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.91500000000000004 or -0.650000000000000022 < (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 92.9%

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

   2. Taylor expanded in l around 0 39.9%

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

   3. Taylor expanded in K around 0 54.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. fma-def 54.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]

      2. associate-*r* 59.0%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot J\right)}\right) + U \]

      3. unpow2 59.0%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\color{blue}{\left(K \cdot K\right)} \cdot \ell\right) \cdot J\right)\right) + U \]

   5. Simplified 59.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right)\right)} + U \]

   6. Taylor expanded in K around inf 64.5%

      \[\leadsto \color{blue}{-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 64.5%

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

      2. associate-*r* 64.5%

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

      3. associate-*r* 69.5%

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

      4. associate-*r* 69.5%

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

      5. unpow2 69.5%

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

      6. *-commutative 69.5%

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

      7. unpow2 69.5%

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

      8. *-commutative 69.5%

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

      9. associate-*r* 74.6%

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

   8. Simplified 74.6%

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

   if -0.91500000000000004 < (cos.f64 (/.f64 K 2)) < -0.650000000000000022

   1. Initial program 84.0%

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

   2. Taylor expanded in l around 0 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 89.0%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 89.0%

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

      2. *-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*l* 89.0%

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

      5. *-commutative 89.0%

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

   5. Simplified 89.0%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0 86.5%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
   3. Step-by-step derivation

      1. expm1-log1p-u 64.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 63.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. sinh-undef 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot J\right)} - 1\right) + U \]

   4. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 69.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)\right)} + U \]

      2. expm1-log1p 96.8%

         \[\leadsto \color{blue}{\left(2 \cdot \sinh \ell\right) \cdot J} + U \]

      3. *-commutative 96.8%

         \[\leadsto \color{blue}{\left(\sinh \ell \cdot 2\right)} \cdot J + U \]

      4. associate-*l* 96.8%

         \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]

   6. Simplified 96.8%

      \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.915:\\ \;\;\;\;U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.65:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 4: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.68:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.68)
     (+ U (* t_0 (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))
     (+ U (* (sinh l) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.68) {
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))));
	} else {
		tmp = U + (sinh(l) * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.68d0) then
        tmp = u + (t_0 * (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
    else
        tmp = u + (sinh(l) * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.68) {
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))));
	} else {
		tmp = U + (Math.sinh(l) * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.68:
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))))
	else:
		tmp = U + (math.sinh(l) * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.68)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0))))));
	else
		tmp = Float64(U + Float64(sinh(l) * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.68)
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))));
	else
		tmp = U + (sinh(l) * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.68], N[(U + N[(t$95$0 * N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.680000000000000049

   1. Initial program 87.7%

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

   2. Taylor expanded in l around 0 94.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.680000000000000049 < (cos.f64 (/.f64 K 2))

   1. Initial program 87.1%

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

   2. Taylor expanded in K around 0 87.1%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
   3. Step-by-step derivation

      1. expm1-log1p-u 63.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 63.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. sinh-undef 66.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot J\right)} - 1\right) + U \]

   4. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 70.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)\right)} + U \]

      2. expm1-log1p 99.3%

         \[\leadsto \color{blue}{\left(2 \cdot \sinh \ell\right) \cdot J} + U \]

      3. *-commutative 99.3%

         \[\leadsto \color{blue}{\left(\sinh \ell \cdot 2\right)} \cdot J + U \]

      4. associate-*l* 99.3%

         \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]

   6. Simplified 99.3%

      \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.68:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.03:\\ \;\;\;\;U + t_0 \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.03)
     (+ U (* t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
     (+ U (* (sinh l) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.03) {
		tmp = U + (t_0 * (pow(l, 3.0) * (J * 0.3333333333333333)));
	} else {
		tmp = U + (sinh(l) * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.03d0) then
        tmp = u + (t_0 * ((l ** 3.0d0) * (j * 0.3333333333333333d0)))
    else
        tmp = u + (sinh(l) * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.03) {
		tmp = U + (t_0 * (Math.pow(l, 3.0) * (J * 0.3333333333333333)));
	} else {
		tmp = U + (Math.sinh(l) * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.03:
		tmp = U + (t_0 * (math.pow(l, 3.0) * (J * 0.3333333333333333)))
	else:
		tmp = U + (math.sinh(l) * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.03)
		tmp = Float64(U + Float64(t_0 * Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))));
	else
		tmp = Float64(U + Float64(sinh(l) * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.03)
		tmp = U + (t_0 * ((l ^ 3.0) * (J * 0.3333333333333333)));
	else
		tmp = U + (sinh(l) * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.03], N[(U + N[(t$95$0 * N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.029999999999999999

   1. Initial program 90.2%

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

   2. Taylor expanded in l around 0 93.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 83.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 83.4%

         \[\leadsto \color{blue}{\left(\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 83.4%

         \[\leadsto \left(\color{blue}{\left({\ell}^{3} \cdot 0.3333333333333333\right)} \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 83.4%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(0.3333333333333333 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(0.3333333333333333 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.029999999999999999 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0 86.5%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
   3. Step-by-step derivation

      1. expm1-log1p-u 64.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 63.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. sinh-undef 66.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot J\right)} - 1\right) + U \]

   4. Applied egg-rr 66.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 69.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)\right)} + U \]

      2. expm1-log1p 96.9%

         \[\leadsto \color{blue}{\left(2 \cdot \sinh \ell\right) \cdot J} + U \]

      3. *-commutative 96.9%

         \[\leadsto \color{blue}{\left(\sinh \ell \cdot 2\right)} \cdot J + U \]

      4. associate-*l* 96.9%

         \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]

   6. Simplified 96.9%

      \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.03:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 6: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ U (* J (* -0.25 (* K (* l K)))))
   (+ U (* (sinh l) (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = U + (J * (-0.25 * (K * (l * K))));
	} else {
		tmp = U + (sinh(l) * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.01d0)) then
        tmp = u + (j * ((-0.25d0) * (k * (l * k))))
    else
        tmp = u + (sinh(l) * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.01) {
		tmp = U + (J * (-0.25 * (K * (l * K))));
	} else {
		tmp = U + (Math.sinh(l) * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.01:
		tmp = U + (J * (-0.25 * (K * (l * K))))
	else:
		tmp = U + (math.sinh(l) * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U + Float64(J * Float64(-0.25 * Float64(K * Float64(l * K)))));
	else
		tmp = Float64(U + Float64(sinh(l) * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.01)
		tmp = U + (J * (-0.25 * (K * (l * K))));
	else
		tmp = U + (sinh(l) * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(J * N[(-0.25 * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 90.1%

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

   2. Taylor expanded in l around 0 55.7%

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

   3. Taylor expanded in K around 0 47.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. fma-def 47.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]

      2. associate-*r* 50.9%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot J\right)}\right) + U \]

      3. unpow2 50.9%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\color{blue}{\left(K \cdot K\right)} \cdot \ell\right) \cdot J\right)\right) + U \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right)\right)} + U \]

   6. Taylor expanded in K around inf 58.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 58.2%

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

      2. associate-*r* 58.2%

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

      3. associate-*r* 61.6%

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

      4. associate-*r* 61.6%

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

      5. unpow2 61.6%

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

      6. *-commutative 61.6%

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

      7. unpow2 61.6%

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

      8. *-commutative 61.6%

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

      9. associate-*r* 67.0%

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

   8. Simplified 67.0%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.5%

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

   2. Taylor expanded in K around 0 86.5%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
   3. Step-by-step derivation

      1. expm1-log1p-u 64.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)} + U \]

      2. expm1-udef 63.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} - 1\right)} + U \]

      3. sinh-undef 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot J\right)} - 1\right) + U \]

   4. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)} - 1\right)} + U \]
   5. Step-by-step derivation

      1. expm1-def 69.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \sinh \ell\right) \cdot J\right)\right)} + U \]

      2. expm1-log1p 96.8%

         \[\leadsto \color{blue}{\left(2 \cdot \sinh \ell\right) \cdot J} + U \]

      3. *-commutative 96.8%

         \[\leadsto \color{blue}{\left(\sinh \ell \cdot 2\right)} \cdot J + U \]

      4. associate-*l* 96.8%

         \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]

   6. Simplified 96.8%

      \[\leadsto \color{blue}{\sinh \ell \cdot \left(2 \cdot J\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 7: 60.0% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+31} \lor \neg \left(\ell \leq 1.08 \cdot 10^{+51}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.9e+31) (not (<= l 1.08e+51)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.9e+31) || !(l <= 1.08e+51)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-5.9d+31)) .or. (.not. (l <= 1.08d+51))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.9e+31) || !(l <= 1.08e+51)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.9e+31) or not (l <= 1.08e+51):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.9e+31) || !(l <= 1.08e+51))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -5.9e+31) || ~((l <= 1.08e+51)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.9e+31], N[Not[LessEqual[l, 1.08e+51]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.9000000000000004e31 or 1.08e51 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 27.5%

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

   3. Taylor expanded in K around 0 19.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. +-commutative 19.4%

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

      2. associate-*r* 19.4%

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

      3. distribute-rgt-out 36.6%

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

      4. *-commutative 36.6%

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

      5. unpow2 36.6%

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

   5. Simplified 36.6%

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

   if -5.9000000000000004e31 < l < 1.08e51

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 90.7%

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

   3. Taylor expanded in K around 0 60.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. fma-def 60.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]

      2. associate-*r* 60.5%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot J\right)}\right) + U \]

      3. unpow2 60.5%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\color{blue}{\left(K \cdot K\right)} \cdot \ell\right) \cdot J\right)\right) + U \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right)\right)} + U \]

   6. Taylor expanded in K around 0 81.9%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* 81.9%

         \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot J} + U \]

      2. *-commutative 81.9%

         \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

   8. Simplified 81.9%

      \[\leadsto \color{blue}{\left(\ell \cdot 2\right) \cdot J} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+31} \lor \neg \left(\ell \leq 1.08 \cdot 10^{+51}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 53.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 1.7 \cdot 10^{-176} \lor \neg \left(J \leq 2 \cdot 10^{+58}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J 1.7e-176) (not (<= J 2e+58)))
   (+ U (* J (* l 2.0)))
   (+ U (* J (* -0.25 (* K (* l K)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= 1.7e-176) || !(J <= 2e+58)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (J * (-0.25 * (K * (l * K))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((j <= 1.7d-176) .or. (.not. (j <= 2d+58))) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = u + (j * ((-0.25d0) * (k * (l * k))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= 1.7e-176) || !(J <= 2e+58)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (J * (-0.25 * (K * (l * K))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= 1.7e-176) or not (J <= 2e+58):
		tmp = U + (J * (l * 2.0))
	else:
		tmp = U + (J * (-0.25 * (K * (l * K))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= 1.7e-176) || !(J <= 2e+58))
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(U + Float64(J * Float64(-0.25 * Float64(K * Float64(l * K)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= 1.7e-176) || ~((J <= 2e+58)))
		tmp = U + (J * (l * 2.0));
	else
		tmp = U + (J * (-0.25 * (K * (l * K))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, 1.7e-176], N[Not[LessEqual[J, 2e+58]], $MachinePrecision]], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(-0.25 * N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.6999999999999999e-176 or 1.99999999999999989e58 < J

   1. Initial program 85.8%

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

   2. Taylor expanded in l around 0 67.6%

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

   3. Taylor expanded in K around 0 39.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. fma-def 39.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]

      2. associate-*r* 46.2%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot J\right)}\right) + U \]

      3. unpow2 46.2%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\color{blue}{\left(K \cdot K\right)} \cdot \ell\right) \cdot J\right)\right) + U \]

   5. Simplified 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right)\right)} + U \]

   6. Taylor expanded in K around 0 58.9%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* 58.9%

         \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot J} + U \]

      2. *-commutative 58.9%

         \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

   8. Simplified 58.9%

      \[\leadsto \color{blue}{\left(\ell \cdot 2\right) \cdot J} + U \]

   if 1.6999999999999999e-176 < J < 1.99999999999999989e58

   1. Initial program 94.0%

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

   2. Taylor expanded in l around 0 38.3%

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

   3. Taylor expanded in K around 0 52.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. fma-def 52.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]

      2. associate-*r* 54.0%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot J\right)}\right) + U \]

      3. unpow2 54.0%

         \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\color{blue}{\left(K \cdot K\right)} \cdot \ell\right) \cdot J\right)\right) + U \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right)\right)} + U \]

   6. Taylor expanded in K around inf 49.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 49.0%

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

      2. associate-*r* 49.0%

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

      3. associate-*r* 50.8%

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

      4. associate-*r* 50.8%

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

      5. unpow2 50.8%

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

      6. *-commutative 50.8%

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

      7. unpow2 50.8%

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

      8. *-commutative 50.8%

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

      9. associate-*r* 50.8%

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

   8. Simplified 50.8%

      \[\leadsto \color{blue}{J \cdot \left(-0.25 \cdot \left(\left(\ell \cdot K\right) \cdot K\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.7 \cdot 10^{-176} \lor \neg \left(J \leq 2 \cdot 10^{+58}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 9: 45.1% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -32 \lor \neg \left(\ell \leq 1.2 \cdot 10^{-16}\right):\\ \;\;\;\;U + J \cdot \left(K \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -32.0) (not (<= l 1.2e-16))) (+ U (* J (* K K))) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -32.0) || !(l <= 1.2e-16)) {
		tmp = U + (J * (K * K));
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-32.0d0)) .or. (.not. (l <= 1.2d-16))) then
        tmp = u + (j * (k * k))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -32.0) || !(l <= 1.2e-16)) {
		tmp = U + (J * (K * K));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -32.0) or not (l <= 1.2e-16):
		tmp = U + (J * (K * K))
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -32.0) || !(l <= 1.2e-16))
		tmp = Float64(U + Float64(J * Float64(K * K)));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -32.0) || ~((l <= 1.2e-16)))
		tmp = U + (J * (K * K));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -32.0], N[Not[LessEqual[l, 1.2e-16]], $MachinePrecision]], N[(U + N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -32 or 1.20000000000000002e-16 < l

   1. Initial program 99.4%

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

   3. Applied egg-rr 5.0%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in K around 0 16.1%

      \[\leadsto \color{blue}{\left({K}^{2} \cdot J + -8 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. distribute-rgt-out 16.1%

         \[\leadsto \color{blue}{J \cdot \left({K}^{2} + -8\right)} + U \]

      2. unpow2 16.1%

         \[\leadsto J \cdot \left(\color{blue}{K \cdot K} + -8\right) + U \]

   6. Simplified 16.1%

      \[\leadsto \color{blue}{J \cdot \left(K \cdot K + -8\right)} + U \]

   7. Taylor expanded in K around inf 16.1%

      \[\leadsto \color{blue}{{K}^{2} \cdot J} + U \]
   8. Step-by-step derivation

      1. *-commutative 16.1%

         \[\leadsto \color{blue}{J \cdot {K}^{2}} + U \]

      2. unpow2 16.1%

         \[\leadsto J \cdot \color{blue}{\left(K \cdot K\right)} + U \]

   9. Simplified 16.1%

      \[\leadsto \color{blue}{J \cdot \left(K \cdot K\right)} + U \]

   if -32 < l < 1.20000000000000002e-16

   1. Initial program 73.6%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \left(J \cdot \color{blue}{0.125}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in J around 0 73.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -32 \lor \neg \left(\ell \leq 1.2 \cdot 10^{-16}\right):\\ \;\;\;\;U + J \cdot \left(K \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 10: 54.9% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot 2\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J (* l 2.0))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * 2.0));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * 2.0d0))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * 2.0));
}

Python

def code(J, l, K, U):
	return U + (J * (l * 2.0))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * 2.0)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * 2.0));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.3%

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

2. Taylor expanded in l around 0 62.1%

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

3. Taylor expanded in K around 0 41.8%

   \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. fma-def 41.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]

   2. associate-*r* 47.6%

      \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \color{blue}{\left(\left({K}^{2} \cdot \ell\right) \cdot J\right)}\right) + U \]

   3. unpow2 47.6%

      \[\leadsto \mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\color{blue}{\left(K \cdot K\right)} \cdot \ell\right) \cdot J\right)\right) + U \]

5. Simplified 47.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \ell \cdot J, -0.25 \cdot \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right)\right)} + U \]

6. Taylor expanded in K around 0 54.2%

   \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} + U \]
7. Step-by-step derivation

   1. associate-*r* 54.2%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot J} + U \]

   2. *-commutative 54.2%

      \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

8. Simplified 54.2%

   \[\leadsto \color{blue}{\left(\ell \cdot 2\right) \cdot J} + U \]

9. Final simplification 54.2%

   \[\leadsto U + J \cdot \left(\ell \cdot 2\right) \]

Alternative 11: 37.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 87.3%

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

3. Applied egg-rr 27.8%

   \[\leadsto \left(J \cdot \color{blue}{0.125}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Taylor expanded in J around 0 37.0%

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

5. Final simplification 37.0%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023187 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))