Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.5%

Time: 11.9s

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.

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

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right) + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

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 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+
      U
      (+
       (* (pow l 3.0) (* J 0.3333333333333333))
       (* (* J (cos (* K 0.5))) (* 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 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + ((pow(l, 3.0) * (J * 0.3333333333333333)) + ((J * cos((K * 0.5))) * (l * 2.0)));
	}
	return tmp;
}

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.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0)) {
		tmp = (Math.cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + ((Math.pow(l, 3.0) * (J * 0.3333333333333333)) + ((J * Math.cos((K * 0.5))) * (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.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0):
		tmp = (math.cos((K / 2.0)) * (t_0 * J)) + U
	else:
		tmp = U + ((math.pow(l, 3.0) * (J * 0.3333333333333333)) + ((J * math.cos((K * 0.5))) * (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 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.0))
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)) + U);
	else
		tmp = Float64(U + Float64(Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)) + Float64(Float64(J * cos(Float64(K * 0.5))) * 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 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0)))
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	else
		tmp = U + (((l ^ 3.0) * (J * 0.3333333333333333)) + ((J * cos((K * 0.5))) * (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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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))) < -inf.0 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 -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 74.8%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]
      Step-by-step derivation

      [Start] 99.9	\[ \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
      associate-*r* [=>] 99.9	\[ \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
      associate-*r* [=>] 99.9	\[ \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]
      associate-*r* [=>] 99.9	\[ \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]
      associate-*r* [=>] 99.9	\[ \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]
      distribute-rgt-out [=>] 99.9	\[ \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]
      *-commutative [=>] 99.9	\[ J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]
      associate-*r* [=>] 99.9	\[ J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]
      *-commutative [=>] 99.9	\[ J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]
      associate-*r* [=>] 99.9	\[ J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]
      distribute-rgt-out [=>] 99.9	\[ J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]
      +-commutative [<=] 99.9	\[ J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]
      fma-def [=>] 99.9	\[ J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right)} + U \]
      Step-by-step derivation

      [Start] 99.9	\[ J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right) + U \]
      associate-*r* [=>] 100.0	\[ \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)} + U \]
      fma-udef [=>] 100.0	\[ \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)} + U \]
      distribute-lft-in [=>] 100.0	\[ \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 \cdot \ell\right)\right)} + U \]
      *-commutative [=>] 100.0	\[ \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) + U \]

   5. Taylor expanded in K around 0 100.0%

      \[\leadsto \left(\color{blue}{0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)} + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right) + U \]

   6. Simplified 100.0%

      \[\leadsto \left(\color{blue}{{\ell}^{3} \cdot \left(0.3333333333333333 \cdot J\right)} + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right) + U \]
      Step-by-step derivation

      [Start] 100.0	\[ \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right) + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right) + U \]
      associate-*r* [=>] 100.0	\[ \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot J} + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right) + U \]
      *-commutative [=>] 100.0	\[ \left(\color{blue}{\left({\ell}^{3} \cdot 0.3333333333333333\right)} \cdot J + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right) + U \]
      associate-*l* [=>] 100.0	\[ \left(\color{blue}{{\ell}^{3} \cdot \left(0.3333333333333333 \cdot J\right)} + \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot 2\right)\right) + U \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right) + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	93.8%
Cost	14348

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(0.3333333333333333 \cdot t_0\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.013:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-45}:\\ \;\;\;\;U + \left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right) + \left(J \cdot t_0\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	93.8%
Cost	14096

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(0.3333333333333333 \cdot t_0\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.00028:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-45}:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+84}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	78.7%
Cost	13828

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.05:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \]

Alternative 4
Accuracy	81.7%
Cost	13648

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ t_1 := U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.42 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -490:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	86.1%
Cost	13577

\[\begin{array}{l} \mathbf{if}\;\ell \leq -0.015 \lor \neg \left(\ell \leq 1.1 \cdot 10^{-45}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	65.7%
Cost	7633

\[\begin{array}{l} t_0 := U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{if}\;\ell \leq -2.26 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -620:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{elif}\;\ell \leq 2.65 \cdot 10^{+17} \lor \neg \left(\ell \leq 2.45 \cdot 10^{+201}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \]

Alternative 7
Accuracy	47.0%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{elif}\;\ell \leq 2.65 \cdot 10^{+17}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \]

Alternative 8
Accuracy	47.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-13} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Accuracy	43.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.46 \cdot 10^{-30} \lor \neg \left(\ell \leq 1.95 \cdot 10^{+19}\right):\\ \;\;\;\;U + J \cdot \left(-0.5 + \left(K \cdot K\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 10
Accuracy	39.3%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 11
Accuracy	2.7%
Cost	64

\[1 \]

Alternative 12
Accuracy	37.3%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023162 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))