Logistic function from Lakshay Garg

Average Error: 29.6 → 0.2

Time: 8.5s

Precision: binary64

Cost: 46216

Math

\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t_0}, {t_0}^{0.6666666666666666}, -1\right)\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (pow (exp x) -2.0)))))
   (if (<= (* -2.0 x) -200.0)
     (+ (/ 2.0 (+ 1.0 (exp (/ x -0.5)))) -1.0)
     (if (<= (* -2.0 x) 0.0002)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (fma (cbrt t_0) (pow t_0 0.6666666666666666) -1.0)))))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

↓

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + pow(exp(x), -2.0));
	double tmp;
	if ((-2.0 * x) <= -200.0) {
		tmp = (2.0 / (1.0 + exp((x / -0.5)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = fma(cbrt(t_0), pow(t_0, 0.6666666666666666), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

↓

function code(x, y)
	t_0 = Float64(2.0 / Float64(1.0 + (exp(x) ^ -2.0)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -200.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(x / -0.5)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = fma(cbrt(t_0), (t_0 ^ 0.6666666666666666), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -200.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(x / -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 1/3], $MachinePrecision] * N[Power[t$95$0, 0.6666666666666666], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -200

   1. Initial program 0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Simplified 0

      \[\leadsto \color{blue}{\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1} \]
      Proof
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (/.f64 x -1/2)))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (/.f64 x (Rewrite<= metadata-eval (/.f64 1 -2)))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x -2) 1))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x 1) -2))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 (Rewrite=> /-rgt-identity_binary64 x) -2)))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 x))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) (Rewrite<= metadata-eval (neg.f64 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) 1)): 0 points increase in error, 0 points decrease in error

   if -200 < (*.f64 -2 x) < 2.0000000000000001e-4

   1. Initial program 58.9

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Simplified 58.9

      \[\leadsto \color{blue}{\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1} \]
      Proof
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (/.f64 x -1/2)))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (/.f64 x (Rewrite<= metadata-eval (/.f64 1 -2)))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x -2) 1))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x 1) -2))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 (Rewrite=> /-rgt-identity_binary64 x) -2)))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 x))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) (Rewrite<= metadata-eval (neg.f64 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) 1)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + x} \]

   if 2.0000000000000001e-4 < (*.f64 -2 x)

   1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1} \]
      Proof
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (/.f64 x -1/2)))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (/.f64 x (Rewrite<= metadata-eval (/.f64 1 -2)))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x -2) 1))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x 1) -2))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 (Rewrite=> /-rgt-identity_binary64 x) -2)))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 x))))) -1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) (Rewrite<= metadata-eval (neg.f64 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) 1)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \sqrt[3]{\frac{4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}, -1\right)} \]

   4. Taylor expanded in x around inf 0.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \sqrt[3]{\frac{4}{\color{blue}{{\left(1 + {\left(e^{-x}\right)}^{2}\right)}^{2}}}}, -1\right) \]

   5. Simplified 0.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \sqrt[3]{\frac{4}{\color{blue}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}}, -1\right) \]
      Proof
      (pow.f64 (+.f64 1 (exp.f64 (*.f64 x -2))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (exp.f64 (*.f64 x (Rewrite<= metadata-eval (+.f64 -1 -1))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (exp.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 x) (*.f64 -1 x))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (exp.f64 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 x)) (*.f64 -1 x)))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (exp.f64 (+.f64 (neg.f64 x) (Rewrite<= neg-mul-1_binary64 (neg.f64 x))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (exp.f64 (Rewrite=> distribute-neg-out_binary64 (neg.f64 (+.f64 x x))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (Rewrite<= rec-exp_binary64 (/.f64 1 (exp.f64 (+.f64 x x))))) 2): 1 points increase in error, 2 points decrease in error
      (pow.f64 (+.f64 1 (/.f64 1 (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 x) (exp.f64 x))))) 2): 2 points increase in error, 1 points decrease in error
      (pow.f64 (+.f64 1 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (exp.f64 x) 2)))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (/.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (exp.f64 x) (exp.f64 x))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (/.f64 1 (Rewrite=> prod-exp_binary64 (exp.f64 (+.f64 x x))))) 2): 1 points increase in error, 2 points decrease in error
      (pow.f64 (+.f64 1 (Rewrite=> rec-exp_binary64 (exp.f64 (neg.f64 (+.f64 x x))))) 2): 2 points increase in error, 1 points decrease in error
      (pow.f64 (+.f64 1 (exp.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 x) (neg.f64 x))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 (neg.f64 x)) (exp.f64 (neg.f64 x))))) 2): 0 points increase in error, 2 points decrease in error
      (pow.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (exp.f64 (neg.f64 x)) 2))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (pow.f64 (exp.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))) 2)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (+.f64 1 (pow.f64 (exp.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) 2)) 2): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \color{blue}{{\left(\frac{2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{0.6666666666666666}}, -1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, {\left(\frac{2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{0.6666666666666666}, -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	26440

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.5:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	13.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8114724.489725059:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.353595432915118 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8114724.489725059:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.353595432915118 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 6
Error	43.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -5.319024344387354 \cdot 10^{-296}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	59.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))