Hyperbolic tangent

Percentage Accurate: 9.0% → 99.6%

Time: 5.9s

Alternatives: 7

Speedup: 68.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t\_0}{e^{x} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 9.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t\_0}{e^{x} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{e^{x\_m} - t\_0}{e^{x\_m} + t\_0} \leq 0.6:\\ \;\;\;\;x\_m + \left(-0.3333333333333333 \cdot {x\_m}^{3} + \left(-0.05396825396825397 \cdot {x\_m}^{7} + 0.13333333333333333 \cdot {x\_m}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))))
   (*
    x_s
    (if (<= (/ (- (exp x_m) t_0) (+ (exp x_m) t_0)) 0.6)
      (+
       x_m
       (+
        (* -0.3333333333333333 (pow x_m 3.0))
        (+
         (* -0.05396825396825397 (pow x_m 7.0))
         (* 0.13333333333333333 (pow x_m 5.0)))))
      1.0))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = exp(-x_m);
	double tmp;
	if (((exp(x_m) - t_0) / (exp(x_m) + t_0)) <= 0.6) {
		tmp = x_m + ((-0.3333333333333333 * pow(x_m, 3.0)) + ((-0.05396825396825397 * pow(x_m, 7.0)) + (0.13333333333333333 * pow(x_m, 5.0))));
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x_m)
    if (((exp(x_m) - t_0) / (exp(x_m) + t_0)) <= 0.6d0) then
        tmp = x_m + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (((-0.05396825396825397d0) * (x_m ** 7.0d0)) + (0.13333333333333333d0 * (x_m ** 5.0d0))))
    else
        tmp = 1.0d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.exp(-x_m);
	double tmp;
	if (((Math.exp(x_m) - t_0) / (Math.exp(x_m) + t_0)) <= 0.6) {
		tmp = x_m + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + ((-0.05396825396825397 * Math.pow(x_m, 7.0)) + (0.13333333333333333 * Math.pow(x_m, 5.0))));
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.exp(-x_m)
	tmp = 0
	if ((math.exp(x_m) - t_0) / (math.exp(x_m) + t_0)) <= 0.6:
		tmp = x_m + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + ((-0.05396825396825397 * math.pow(x_m, 7.0)) + (0.13333333333333333 * math.pow(x_m, 5.0))))
	else:
		tmp = 1.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = exp(Float64(-x_m))
	tmp = 0.0
	if (Float64(Float64(exp(x_m) - t_0) / Float64(exp(x_m) + t_0)) <= 0.6)
		tmp = Float64(x_m + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(Float64(-0.05396825396825397 * (x_m ^ 7.0)) + Float64(0.13333333333333333 * (x_m ^ 5.0)))));
	else
		tmp = 1.0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = exp(-x_m);
	tmp = 0.0;
	if (((exp(x_m) - t_0) / (exp(x_m) + t_0)) <= 0.6)
		tmp = x_m + ((-0.3333333333333333 * (x_m ^ 3.0)) + ((-0.05396825396825397 * (x_m ^ 7.0)) + (0.13333333333333333 * (x_m ^ 5.0))));
	else
		tmp = 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[Exp[x$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 0.6], N[(x$95$m + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.05396825396825397 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 0.599999999999999978

   1. Initial program 7.5%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)} \]

   if 0.599999999999999978 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 33.3%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.7%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around inf 5.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 5.7%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      2. *-commutative 5.7%

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      3. cube-mult 5.7%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      4. associate-*l* 5.7%

         \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      5. associate-*l/ 5.7%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      6. *-commutative 5.7%

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      7. distribute-rgt-in 5.7%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

      8. associate-*l* 5.7%

         \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

   6. Simplified 5.7%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

   8. Applied egg-rr 6.7%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

   10. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 0.6:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;x\_m + \left(-0.3333333333333333 \cdot {x\_m}^{3} + {x\_m}^{5} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.25)
    (+ x_m (+ (* -0.3333333333333333 (pow x_m 3.0)) (* (pow x_m 5.0) 0.125)))
    1.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = x_m + ((-0.3333333333333333 * pow(x_m, 3.0)) + (pow(x_m, 5.0) * 0.125));
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.25d0) then
        tmp = x_m + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + ((x_m ** 5.0d0) * 0.125d0))
    else
        tmp = 1.0d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = x_m + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (Math.pow(x_m, 5.0) * 0.125));
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.25:
		tmp = x_m + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (math.pow(x_m, 5.0) * 0.125))
	else:
		tmp = 1.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(x_m + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64((x_m ^ 5.0) * 0.125)));
	else
		tmp = 1.0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = x_m + ((-0.3333333333333333 * (x_m ^ 3.0)) + ((x_m ^ 5.0) * 0.125));
	else
		tmp = 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.25], N[(x$95$m + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 5.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 7.4%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.2%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.125 \cdot {x}^{5}\right)} \]

   if 1.25 < x

   1. Initial program 50.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 7.0%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around inf 7.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 7.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      2. *-commutative 7.0%

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      3. cube-mult 7.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      4. associate-*l* 7.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      5. associate-*l/ 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      6. *-commutative 7.0%

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      7. distribute-rgt-in 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

      8. associate-*l* 7.0%

         \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

   8. Applied egg-rr 8.4%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

   10. Applied egg-rr 88.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + {x}^{5} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;x\_m + \left(-0.3333333333333333 \cdot {x\_m}^{3} + 0.13333333333333333 \cdot {x\_m}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.25)
    (+
     x_m
     (+
      (* -0.3333333333333333 (pow x_m 3.0))
      (* 0.13333333333333333 (pow x_m 5.0))))
    1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.25], N[(x$95$m + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 7.4%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

   if 1.25 < x

   1. Initial program 50.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 7.0%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around inf 7.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 7.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      2. *-commutative 7.0%

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      3. cube-mult 7.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      4. associate-*l* 7.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      5. associate-*l/ 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      6. *-commutative 7.0%

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      7. distribute-rgt-in 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

      8. associate-*l* 7.0%

         \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

   8. Applied egg-rr 8.4%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

   10. Applied egg-rr 88.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 4: 99.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m + -0.3333333333333333 \cdot {x\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.15) (+ x_m (* -0.3333333333333333 (pow x_m 3.0))) 1.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = x_m + (-0.3333333333333333 * pow(x_m, 3.0));
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.15d0) then
        tmp = x_m + ((-0.3333333333333333d0) * (x_m ** 3.0d0))
    else
        tmp = 1.0d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = x_m + (-0.3333333333333333 * Math.pow(x_m, 3.0));
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.15:
		tmp = x_m + (-0.3333333333333333 * math.pow(x_m, 3.0))
	else:
		tmp = 1.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(x_m + Float64(-0.3333333333333333 * (x_m ^ 3.0)));
	else
		tmp = 1.0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.15)
		tmp = x_m + (-0.3333333333333333 * (x_m ^ 3.0));
	else
		tmp = 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.15], N[(x$95$m + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 7.4%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
   4. Step-by-step derivation

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if 1.1499999999999999 < x

   1. Initial program 50.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 7.0%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around inf 7.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 7.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      2. *-commutative 7.0%

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      3. cube-mult 7.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      4. associate-*l* 7.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      5. associate-*l/ 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      6. *-commutative 7.0%

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      7. distribute-rgt-in 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

      8. associate-*l* 7.0%

         \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

   8. Applied egg-rr 8.4%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

   10. Applied egg-rr 88.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 68.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (if (<= x_m 1.0) x_m 1.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = x_m
    else
        tmp = 1.0d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = x_m
	else:
		tmp = 1.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], x$95$m, 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 7.4%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{x} \]

   if 1 < x

   1. Initial program 50.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 7.0%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around inf 7.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 7.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      2. *-commutative 7.0%

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      3. cube-mult 7.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      4. associate-*l* 7.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      5. associate-*l/ 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      6. *-commutative 7.0%

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

      7. distribute-rgt-in 7.0%

         \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

      8. associate-*l* 7.0%

         \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

   8. Applied egg-rr 8.4%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

   10. Applied egg-rr 88.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 5.7% accurate, 409.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.5))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.5;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * 0.5d0
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * 0.5;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.5

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.5)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.5;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.4%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 96.0%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around inf 96.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 96.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   2. *-commutative 96.0%

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   3. cube-mult 96.0%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   4. associate-*l* 96.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   5. associate-*l/ 96.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   6. *-commutative 96.0%

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   7. distribute-rgt-in 96.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

   8. associate-*l* 96.0%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

6. Simplified 96.0%

   \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

8. Applied egg-rr 95.3%

   \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

10. Applied egg-rr 4.0%

    \[\leadsto \color{blue}{0.5} \]

11. Final simplification 4.0%

    \[\leadsto 0.5 \]
12. Add Preprocessing

Alternative 7: 7.4% accurate, 409.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 1.0))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 1.0;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * 1.0d0
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * 1.0;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 1.0

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 1.0)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 8.4%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 96.0%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around inf 96.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{3}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 96.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot {x}^{3}}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   2. *-commutative 96.0%

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.3333333333333333}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   3. cube-mult 96.0%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.3333333333333333}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   4. associate-*l* 96.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{e^{x} + e^{-x}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   5. associate-*l/ 96.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   6. *-commutative 96.0%

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot \frac{x}{e^{x} + e^{-x}}} + 2 \cdot \frac{x}{e^{x} + e^{-x}} \]

   7. distribute-rgt-in 96.0%

      \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)} \]

   8. associate-*l* 96.0%

      \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right) \]

6. Simplified 96.0%

   \[\leadsto \color{blue}{\frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)} \]

8. Applied egg-rr 95.3%

   \[\leadsto \frac{x}{e^{x} + e^{-x}} \cdot \left(x \cdot \color{blue}{2.5} + 2\right) \]

10. Applied egg-rr 5.6%

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

11. Final simplification 5.6%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))