Hyperbolic tangent

Average Accuracy: 9.2% → 98.4%

Time: 1.8min

Precision: binary64

Cost: 13636

Math

\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   -1.0
   (+
    (* -0.3333333333333333 (pow x 3.0))
    (+ x (* 0.13333333333333333 (pow x 5.0))))))

C

double code(double x) {
	return (exp(x) - exp(-x)) / (exp(x) + exp(-x));
}

↓

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = -1.0;
	} else {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / (exp(x) + exp(-x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = -1.0d0
    else
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / (Math.exp(x) + Math.exp(-x));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = -1.0;
	} else {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / (math.exp(x) + math.exp(-x))

↓

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = -1.0
	else:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	return tmp

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / Float64(exp(x) + exp(Float64(-x))))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = -1.0;
	else
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / (exp(x) + exp(-x));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = -1.0;
	else
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -1.25], -1.0, N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25

   1. Initial program 30.7%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

   2. Taylor expanded in x around 0 3.8%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]

   3. Simplified 3.8%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + x \cdot x}} \]
      Proof

      [Start] 3.8	\[ \frac{e^{x} - e^{-x}}{2 + {x}^{2}} \]
      unpow2 [=>] 3.8	\[ \frac{e^{x} - e^{-x}}{2 + \color{blue}{x \cdot x}} \]

   4. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{-1} \]
      Proof

      [Start] 3.8	\[ \frac{e^{x} - e^{-x}}{2 + x \cdot x} \]

   if -1.25 < x

   1. Initial program 9.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

   2. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)}{2 + x \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x}{2 + x \cdot x}\\ \end{array} \]

Alternative 3
Accuracy	7.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	4.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))