Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.9% → 99.6%

Time: 11.2s

Precision: binary64

Cost: 46217

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

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

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

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

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

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+58} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -1e+58) (not (<= t_1 0.0)))
     (+ (* t_0 (* t_1 J)) U)
     (+ U (* t_0 (* J (* l 2.0)))))))

C

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

↓

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 <= -1e+58) || !(t_1 <= 0.0)) {
		tmp = (t_0 * (t_1 * J)) + 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
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

↓

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 <= (-1d+58)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = (t_0 * (t_1 * j)) + 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) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

↓

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 <= -1e+58) || !(t_1 <= 0.0)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Python

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

↓

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 <= -1e+58) or not (t_1 <= 0.0):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * (J * (l * 2.0)))
	return tmp

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

↓

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 <= -1e+58) || !(t_1 <= 0.0))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

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

↓

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 <= -1e+58) || ~((t_1 <= 0.0)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
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]

↓

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, -1e+58], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $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))) < -9.99999999999999944e57 or 0.0 < (-.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 -9.99999999999999944e57 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 64.3%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \cdot 10^{+58} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.5%
Cost	39560

\[\begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := t_0 \cdot J\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_1\\ \end{array} \]

Alternative 2
Accuracy	86.7%
Cost	39433

\[\begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+58} \lor \neg \left(t_0 \leq 2\right):\\ \;\;\;\;t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	72.4%
Cost	14152

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.92:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell + \ell\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.05:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	78.6%
Cost	13828

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.265:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 5
Accuracy	76.5%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.265:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	76.5%
Cost	13764

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.265:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	70.0%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;J \leq -2.3 \cdot 10^{+229} \lor \neg \left(J \leq 1.3 \cdot 10^{+142}\right):\\ \;\;\;\;J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell + \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	69.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-13} \lor \neg \left(\ell \leq 0.36\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 9
Accuracy	69.8%
Cost	832

\[U + \left(\ell \cdot J\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \]

Alternative 10
Accuracy	42.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;\ell \leq -810:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 11
Accuracy	53.7%
Cost	448

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

Alternative 12
Accuracy	2.7%
Cost	64

\[1 \]

Alternative 13
Accuracy	36.6%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023161 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))