Maksimov and Kolovsky, Equation (4)

?

Percentage Accurate: 86.3% → 99.6%
Time: 11.5s
Precision: binary64
Cost: 46217

?

\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
↓
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -0.02 \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;U + \left(t_1 \cdot J\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
(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 -0.02) (not (<= t_1 0.0)))
     (+ U (* (* t_1 J) t_0))
     (+
      U
      (* t_0 (+ (* 2.0 (* l J)) (* 0.3333333333333333 (* J (pow l 3.0)))))))))
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 <= -0.02) || !(t_1 <= 0.0)) {
		tmp = U + ((t_1 * J) * t_0);
	} else {
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * pow(l, 3.0)))));
	}
	return tmp;
}

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 <= (-0.02d0)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = u + ((t_1 * j) * t_0)
    else
        tmp = u + (t_0 * ((2.0d0 * (l * j)) + (0.3333333333333333d0 * (j * (l ** 3.0d0)))))
    end if
    code = tmp
end function

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 <= -0.02) || !(t_1 <= 0.0)) {
		tmp = U + ((t_1 * J) * t_0);
	} else {
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * Math.pow(l, 3.0)))));
	}
	return tmp;
}

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 <= -0.02) or not (t_1 <= 0.0):
		tmp = U + ((t_1 * J) * t_0)
	else:
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * math.pow(l, 3.0)))))
	return tmp

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 <= -0.02) || !(t_1 <= 0.0))
		tmp = Float64(U + Float64(Float64(t_1 * J) * t_0));
	else
		tmp = Float64(U + Float64(t_0 * Float64(Float64(2.0 * Float64(l * J)) + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))));
	end
	return tmp
end

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 <= -0.02) || ~((t_1 <= 0.0)))
		tmp = U + ((t_1 * J) * t_0);
	else
		tmp = U + (t_0 * ((2.0 * (l * J)) + (0.3333333333333333 * (J * (l ^ 3.0)))));
	end
	tmp_2 = tmp;
end

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, -0.02], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(U + N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\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 -0.02 \lor \neg \left(t_1 \leq 0\right):\\
\;\;\;\;U + \left(t_1 \cdot J\right) \cdot t_0\\

\mathbf{else}:\\
\;\;\;\;U + t_0 \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 15 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0200000000000000004 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

    1. Initial program 99.9%

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

    if -0.0200000000000000004 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

    1. Initial program 70.8%

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

    2. Taylor expanded in l around 0 100.0%

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

  4. Final simplification99.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.02 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.6%
Cost46217
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -0.02 \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;U + \left(t_1 \cdot J\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
Alternative 2
Accuracy97.5%
Cost60740
\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq 0.01:\\ \;\;\;\;\left(2 \cdot \left(t_0 \cdot \left(\ell \cdot J\right)\right) + \left(0.016666666666666666 \cdot \left(t_0 \cdot \left(J \cdot {\ell}^{5}\right)\right) + \left(0.0003968253968253968 \cdot \left(t_0 \cdot \left(J \cdot {\ell}^{7}\right)\right) + 0.3333333333333333 \cdot \left(t_0 \cdot \left(J \cdot {\ell}^{3}\right)\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_1 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
Alternative 3
Accuracy97.5%
Cost40388
\[\begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_0 \cdot J\right) \cdot t_1\\ \end{array} \]
Alternative 4
Accuracy96.5%
Cost33924
\[\begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;U + t_1 \cdot \left(2 \cdot \left(\ell \cdot J\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_0 \cdot J\right) \cdot t_1\\ \end{array} \]
Alternative 5
Accuracy96.5%
Cost33668
\[\begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_0 \cdot J\right) \cdot t_1\\ \end{array} \]
Alternative 6
Accuracy93.8%
Cost20612
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.999995:\\ \;\;\;\;U + t_0 \cdot \left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]
Alternative 7
Accuracy93.8%
Cost14148
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.999995:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]
Alternative 8
Accuracy85.2%
Cost13508
\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(K \cdot \left(K \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]
Alternative 9
Accuracy58.6%
Cost7189
\[\begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -840:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 0.00335:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 10^{+97} \lor \neg \left(\ell \leq 9.5 \cdot 10^{+193}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \]
Alternative 10
Accuracy58.6%
Cost1097
\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+76} \lor \neg \left(\ell \leq 0.00335\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
Alternative 11
Accuracy44.1%
Cost585
\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-28} \lor \neg \left(\ell \leq 1.15 \cdot 10^{-62}\right):\\ \;\;\;\;2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 12
Accuracy41.5%
Cost456
\[\begin{array}{l} \mathbf{if}\;\ell \leq -0.0115:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1020:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]
Alternative 13
Accuracy53.7%
Cost448
\[U + 2 \cdot \left(\ell \cdot J\right) \]
Alternative 14
Accuracy2.7%
Cost64
\[1 \]
Alternative 15
Accuracy36.3%
Cost64
\[U \]

Reproduce?

herbie shell --seed 2023271 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))