Hyperbolic tangent

Percentage Accurate: 9.3% → 99.7%

Time: 17.5s

Alternatives: 9

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: 9.3% 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{e^{x_m} - t_0}{e^{x_m} + t_0} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x_m + -0.3333333333333333 \cdot {x_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{e^{x_m \cdot 2}}} + \frac{-1}{{\left(e^{x_m}\right)}^{2} + 1}\\ \end{array} \end{array} \end{array} \]

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)) 2e-6)
      (+ x_m (* -0.3333333333333333 (pow x_m 3.0)))
      (+
       (/ 1.0 (+ 1.0 (/ 1.0 (exp (* x_m 2.0)))))
       (/ -1.0 (+ (pow (exp x_m) 2.0) 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)) <= 2e-6) {
		tmp = x_m + (-0.3333333333333333 * pow(x_m, 3.0));
	} else {
		tmp = (1.0 / (1.0 + (1.0 / exp((x_m * 2.0))))) + (-1.0 / (pow(exp(x_m), 2.0) + 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)) <= 2d-6) then
        tmp = x_m + ((-0.3333333333333333d0) * (x_m ** 3.0d0))
    else
        tmp = (1.0d0 / (1.0d0 + (1.0d0 / exp((x_m * 2.0d0))))) + ((-1.0d0) / ((exp(x_m) ** 2.0d0) + 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)) <= 2e-6) {
		tmp = x_m + (-0.3333333333333333 * Math.pow(x_m, 3.0));
	} else {
		tmp = (1.0 / (1.0 + (1.0 / Math.exp((x_m * 2.0))))) + (-1.0 / (Math.pow(Math.exp(x_m), 2.0) + 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)) <= 2e-6:
		tmp = x_m + (-0.3333333333333333 * math.pow(x_m, 3.0))
	else:
		tmp = (1.0 / (1.0 + (1.0 / math.exp((x_m * 2.0))))) + (-1.0 / (math.pow(math.exp(x_m), 2.0) + 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)) <= 2e-6)
		tmp = Float64(x_m + Float64(-0.3333333333333333 * (x_m ^ 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(1.0 / exp(Float64(x_m * 2.0))))) + Float64(-1.0 / Float64((exp(x_m) ^ 2.0) + 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)) <= 2e-6)
		tmp = x_m + (-0.3333333333333333 * (x_m ^ 3.0));
	else
		tmp = (1.0 / (1.0 + (1.0 / exp((x_m * 2.0))))) + (-1.0 / ((exp(x_m) ^ 2.0) + 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], 2e-6], N[(x$95$m + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(1.0 / N[Exp[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Power[N[Exp[x$95$m], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $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)))) < 1.99999999999999991e-6

   1. Initial program 7.2%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.2%

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

      2. +-commutative 7.2%

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

      3. div-sub 7.2%

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

   3. Simplified 5.8%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.2%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 18.9%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 18.9%

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

      2. +-commutative 18.9%

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

      3. div-sub 19.3%

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

   3. Simplified 86.7%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. pow-exp 99.3%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   7. Applied egg-rr 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{e^{x \cdot 2}}} + \frac{-1}{{\left(e^{x}\right)}^{2} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \frac{\mathsf{expm1}\left(x_m \cdot 2\right) - \mathsf{expm1}\left(x_m \cdot -2\right)}{\left(-1 - {\left(e^{x_m}\right)}^{2}\right) \cdot \left(-1 - {\left(e^{x_m}\right)}^{-2}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (- (expm1 (* x_m 2.0)) (expm1 (* x_m -2.0)))
   (* (- -1.0 (pow (exp x_m) 2.0)) (- -1.0 (pow (exp x_m) -2.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((expm1((x_m * 2.0)) - expm1((x_m * -2.0))) / ((-1.0 - pow(exp(x_m), 2.0)) * (-1.0 - pow(exp(x_m), -2.0))));
}

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 * ((Math.expm1((x_m * 2.0)) - Math.expm1((x_m * -2.0))) / ((-1.0 - Math.pow(Math.exp(x_m), 2.0)) * (-1.0 - Math.pow(Math.exp(x_m), -2.0))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((math.expm1((x_m * 2.0)) - math.expm1((x_m * -2.0))) / ((-1.0 - math.pow(math.exp(x_m), 2.0)) * (-1.0 - math.pow(math.exp(x_m), -2.0))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(expm1(Float64(x_m * 2.0)) - expm1(Float64(x_m * -2.0))) / Float64(Float64(-1.0 - (exp(x_m) ^ 2.0)) * Float64(-1.0 - (exp(x_m) ^ -2.0)))))
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 * N[(N[(N[(Exp[N[(x$95$m * 2.0), $MachinePrecision]] - 1), $MachinePrecision] - N[(Exp[N[(x$95$m * -2.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - N[Power[N[Exp[x$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[Power[N[Exp[x$95$m], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Step-by-step derivation

   1. remove-double-neg 7.5%

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

   2. +-commutative 7.5%

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

   3. div-sub 7.5%

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

3. Simplified 7.4%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 9.0%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]
6. Step-by-step derivation

   1. frac-2neg 9.0%

      \[\leadsto \color{blue}{\frac{-1}{-\left(1 + \frac{1}{{\left(e^{x}\right)}^{2}}\right)}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   2. metadata-eval 9.0%

      \[\leadsto \frac{\color{blue}{-1}}{-\left(1 + \frac{1}{{\left(e^{x}\right)}^{2}}\right)} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   3. frac-2neg 9.0%

      \[\leadsto \frac{-1}{-\left(1 + \frac{1}{{\left(e^{x}\right)}^{2}}\right)} - \color{blue}{\frac{-1}{-\left(1 + {\left(e^{x}\right)}^{2}\right)}} \]

   4. metadata-eval 9.0%

      \[\leadsto \frac{-1}{-\left(1 + \frac{1}{{\left(e^{x}\right)}^{2}}\right)} - \frac{\color{blue}{-1}}{-\left(1 + {\left(e^{x}\right)}^{2}\right)} \]

   5. frac-sub 7.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-\left(1 + {\left(e^{x}\right)}^{2}\right)\right) - \left(-\left(1 + \frac{1}{{\left(e^{x}\right)}^{2}}\right)\right) \cdot -1}{\left(-\left(1 + \frac{1}{{\left(e^{x}\right)}^{2}}\right)\right) \cdot \left(-\left(1 + {\left(e^{x}\right)}^{2}\right)\right)}} \]

7. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 + \left(-{\left(e^{x}\right)}^{2}\right)\right) - \left(-1 + \left(-e^{-2 \cdot x}\right)\right) \cdot -1}{\left(-1 + \left(-e^{-2 \cdot x}\right)\right) \cdot \left(-1 + \left(-{\left(e^{x}\right)}^{2}\right)\right)}} \]
8. Step-by-step derivation

9. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x \cdot 2\right) - \mathsf{expm1}\left(x \cdot -2\right)}{\left(-1 - {\left(e^{x}\right)}^{2}\right) \cdot \left(-1 - {\left(e^{x}\right)}^{-2}\right)}} \]

10. Final simplification 98.4%

    \[\leadsto \frac{\mathsf{expm1}\left(x \cdot 2\right) - \mathsf{expm1}\left(x \cdot -2\right)}{\left(-1 - {\left(e^{x}\right)}^{2}\right) \cdot \left(-1 - {\left(e^{x}\right)}^{-2}\right)} \]
11. Add Preprocessing

Alternative 3: 99.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x_m \cdot 2}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.00095:\\ \;\;\;\;x_m + -0.3333333333333333 \cdot {x_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{t_0}} + \frac{-1}{1 + t_0}\\ \end{array} \end{array} \end{array} \]

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 2.0))))
   (*
    x_s
    (if (<= x_m 0.00095)
      (+ x_m (* -0.3333333333333333 (pow x_m 3.0)))
      (+ (/ 1.0 (+ 1.0 (/ 1.0 t_0))) (/ -1.0 (+ 1.0 t_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 * 2.0));
	double tmp;
	if (x_m <= 0.00095) {
		tmp = x_m + (-0.3333333333333333 * pow(x_m, 3.0));
	} else {
		tmp = (1.0 / (1.0 + (1.0 / t_0))) + (-1.0 / (1.0 + t_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 * 2.0d0))
    if (x_m <= 0.00095d0) then
        tmp = x_m + ((-0.3333333333333333d0) * (x_m ** 3.0d0))
    else
        tmp = (1.0d0 / (1.0d0 + (1.0d0 / t_0))) + ((-1.0d0) / (1.0d0 + t_0))
    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 * 2.0));
	double tmp;
	if (x_m <= 0.00095) {
		tmp = x_m + (-0.3333333333333333 * Math.pow(x_m, 3.0));
	} else {
		tmp = (1.0 / (1.0 + (1.0 / t_0))) + (-1.0 / (1.0 + t_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 * 2.0))
	tmp = 0
	if x_m <= 0.00095:
		tmp = x_m + (-0.3333333333333333 * math.pow(x_m, 3.0))
	else:
		tmp = (1.0 / (1.0 + (1.0 / t_0))) + (-1.0 / (1.0 + t_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 * 2.0))
	tmp = 0.0
	if (x_m <= 0.00095)
		tmp = Float64(x_m + Float64(-0.3333333333333333 * (x_m ^ 3.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(1.0 / t_0))) + Float64(-1.0 / Float64(1.0 + t_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 * 2.0));
	tmp = 0.0;
	if (x_m <= 0.00095)
		tmp = x_m + (-0.3333333333333333 * (x_m ^ 3.0));
	else
		tmp = (1.0 / (1.0 + (1.0 / t_0))) + (-1.0 / (1.0 + t_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[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.00095], N[(x$95$m + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.49999999999999998e-4

   1. Initial program 7.2%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.2%

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

      2. +-commutative 7.2%

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

      3. div-sub 7.2%

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

   3. Simplified 6.2%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.8%

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

   if 9.49999999999999998e-4 < x

   1. Initial program 23.6%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 23.6%

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

      2. +-commutative 23.6%

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

      3. div-sub 24.1%

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

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. pow-exp 99.1%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   7. Applied egg-rr 99.1%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. pow-exp 99.1%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   9. Applied egg-rr 98.6%

      \[\leadsto \frac{1}{1 + \frac{1}{e^{x \cdot 2}}} - \frac{1}{1 + \color{blue}{e^{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00095:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{e^{x \cdot 2}}} + \frac{-1}{1 + e^{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.15:\\ \;\;\;\;x_m + \left(-0.3333333333333333 \cdot {x_m}^{3} + 0.13333333333333333 \cdot {x_m}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 + e^{x_m \cdot 2}}\\ \end{array} \end{array} \]

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))
      (* 0.13333333333333333 (pow x_m 5.0))))
    (+ 1.0 (/ -1.0 (+ 1.0 (exp (* x_m 2.0))))))))

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)) + (0.13333333333333333 * pow(x_m, 5.0)));
	} else {
		tmp = 1.0 + (-1.0 / (1.0 + exp((x_m * 2.0))));
	}
	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)) + (0.13333333333333333d0 * (x_m ** 5.0d0)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 + exp((x_m * 2.0d0))))
    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)) + (0.13333333333333333 * Math.pow(x_m, 5.0)));
	} else {
		tmp = 1.0 + (-1.0 / (1.0 + Math.exp((x_m * 2.0))));
	}
	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)) + (0.13333333333333333 * math.pow(x_m, 5.0)))
	else:
		tmp = 1.0 + (-1.0 / (1.0 + math.exp((x_m * 2.0))))
	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(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(0.13333333333333333 * (x_m ^ 5.0))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(x_m * 2.0)))));
	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)) + (0.13333333333333333 * (x_m ^ 5.0)));
	else
		tmp = 1.0 + (-1.0 / (1.0 + exp((x_m * 2.0))));
	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[(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], N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 7.6%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.6%

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

      2. +-commutative 7.6%

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

      3. div-sub 7.6%

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

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.8%

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

   if 1.1499999999999999 < x

   1. Initial program 0.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. div-sub 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]

   6. Taylor expanded in x around 0 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + 2 \cdot x}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   8. Simplified 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + x \cdot 2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. pow-exp 100.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{e^{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 + e^{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.15:\\ \;\;\;\;\left(x_m + -0.3333333333333333 \cdot {x_m}^{3}\right) + 0.13333333333333333 \cdot {x_m}^{5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 + e^{x_m \cdot 2}}\\ \end{array} \end{array} \]

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)))
     (* 0.13333333333333333 (pow x_m 5.0)))
    (+ 1.0 (/ -1.0 (+ 1.0 (exp (* x_m 2.0))))))))

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))) + (0.13333333333333333 * pow(x_m, 5.0));
	} else {
		tmp = 1.0 + (-1.0 / (1.0 + exp((x_m * 2.0))));
	}
	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))) + (0.13333333333333333d0 * (x_m ** 5.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 + exp((x_m * 2.0d0))))
    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))) + (0.13333333333333333 * Math.pow(x_m, 5.0));
	} else {
		tmp = 1.0 + (-1.0 / (1.0 + Math.exp((x_m * 2.0))));
	}
	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))) + (0.13333333333333333 * math.pow(x_m, 5.0))
	else:
		tmp = 1.0 + (-1.0 / (1.0 + math.exp((x_m * 2.0))))
	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(Float64(x_m + Float64(-0.3333333333333333 * (x_m ^ 3.0))) + Float64(0.13333333333333333 * (x_m ^ 5.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(x_m * 2.0)))));
	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))) + (0.13333333333333333 * (x_m ^ 5.0));
	else
		tmp = 1.0 + (-1.0 / (1.0 + exp((x_m * 2.0))));
	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[(N[(x$95$m + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 7.6%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.6%

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

      2. +-commutative 7.6%

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

      3. div-sub 7.6%

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

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 98.8%

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

      2. +-commutative 98.8%

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

      3. fma-def 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, x\right) + 0.13333333333333333 \cdot {x}^{5}} \]
   8. Step-by-step derivation

      1. fma-udef 98.8%

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

   9. Applied egg-rr 98.8%

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

   if 1.1499999999999999 < x

   1. Initial program 0.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. div-sub 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]

   6. Taylor expanded in x around 0 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + 2 \cdot x}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   8. Simplified 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + x \cdot 2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. pow-exp 100.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{e^{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\left(x + -0.3333333333333333 \cdot {x}^{3}\right) + 0.13333333333333333 \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 + e^{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.05:\\ \;\;\;\;x_m + -0.3333333333333333 \cdot {x_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 + e^{x_m \cdot 2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.05)
    (+ x_m (* -0.3333333333333333 (pow x_m 3.0)))
    (+ 1.0 (/ -1.0 (+ 1.0 (exp (* x_m 2.0))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = x_m + (-0.3333333333333333 * pow(x_m, 3.0));
	} else {
		tmp = 1.0 + (-1.0 / (1.0 + exp((x_m * 2.0))));
	}
	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 <= 1.05d0) then
        tmp = x_m + ((-0.3333333333333333d0) * (x_m ** 3.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 + exp((x_m * 2.0d0))))
    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 <= 1.05) {
		tmp = x_m + (-0.3333333333333333 * Math.pow(x_m, 3.0));
	} else {
		tmp = 1.0 + (-1.0 / (1.0 + Math.exp((x_m * 2.0))));
	}
	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 <= 1.05:
		tmp = x_m + (-0.3333333333333333 * math.pow(x_m, 3.0))
	else:
		tmp = 1.0 + (-1.0 / (1.0 + math.exp((x_m * 2.0))))
	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 <= 1.05)
		tmp = Float64(x_m + Float64(-0.3333333333333333 * (x_m ^ 3.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(x_m * 2.0)))));
	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 <= 1.05)
		tmp = x_m + (-0.3333333333333333 * (x_m ^ 3.0));
	else
		tmp = 1.0 + (-1.0 / (1.0 + exp((x_m * 2.0))));
	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, 1.05], N[(x$95$m + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 7.6%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.6%

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

      2. +-commutative 7.6%

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

      3. div-sub 7.6%

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

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.6%

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

   if 1.05000000000000004 < x

   1. Initial program 0.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. div-sub 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]

   6. Taylor expanded in x around 0 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + 2 \cdot x}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   8. Simplified 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + x \cdot 2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. pow-exp 100.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{x \cdot 2}}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{e^{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 + e^{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.2:\\ \;\;\;\;x_m + -0.3333333333333333 \cdot {x_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.2) (+ 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 <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		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 <= 1.2:
		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 <= 1.2)
		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 <= 1.2)
		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, 1.2], 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 < 1.19999999999999996

   1. Initial program 7.6%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.6%

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

      2. +-commutative 7.6%

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

      3. div-sub 7.6%

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

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.6%

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

   if 1.19999999999999996 < x

   1. Initial program 0.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. div-sub 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]

   6. Taylor expanded in x around 0 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{\color{blue}{2 + 2 \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 56.2%

         \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{2 + \color{blue}{x \cdot 2}} \]

   8. Simplified 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{\color{blue}{2 + x \cdot 2}} \]

   9. Taylor expanded in x around 0 54.6%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + 2 \cdot x}}} - \frac{1}{2 + x \cdot 2} \]
   10. Step-by-step derivation

       1. *-commutative 54.6%

          \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   11. Simplified 54.6%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   12. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% 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 7.6%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 7.6%

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

      2. +-commutative 7.6%

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

      3. div-sub 7.6%

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

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.6%

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

   if 1 < x

   1. Initial program 0.0%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Step-by-step derivation

      1. remove-double-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. div-sub 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]

   6. Taylor expanded in x around 0 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{\color{blue}{2 + 2 \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 56.2%

         \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{2 + \color{blue}{x \cdot 2}} \]

   8. Simplified 56.2%

      \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{\color{blue}{2 + x \cdot 2}} \]

   9. Taylor expanded in x around 0 54.6%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + 2 \cdot x}}} - \frac{1}{2 + x \cdot 2} \]
   10. Step-by-step derivation

       1. *-commutative 54.6%

          \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   11. Simplified 54.6%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

   12. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.7%

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

Alternative 9: 7.9% 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 7.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Step-by-step derivation

   1. remove-double-neg 7.5%

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

   2. +-commutative 7.5%

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

   3. div-sub 7.5%

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

3. Simplified 7.4%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{-2}\right)\right) - \mathsf{reciprocal}\left(\left(1 + {\left(e^{x}\right)}^{2}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 9.0%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{1 + {\left(e^{x}\right)}^{2}}} \]

6. Taylor expanded in x around 0 7.0%

   \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{\color{blue}{2 + 2 \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 7.0%

      \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{2 + \color{blue}{x \cdot 2}} \]

8. Simplified 7.0%

   \[\leadsto \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} - \frac{1}{\color{blue}{2 + x \cdot 2}} \]

9. Taylor expanded in x around 0 6.9%

   \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + 2 \cdot x}}} - \frac{1}{2 + x \cdot 2} \]
10. Step-by-step derivation

    1. *-commutative 6.9%

       \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

11. Simplified 6.9%

    \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + x \cdot 2}}} - \frac{1}{2 + x \cdot 2} \]

12. Taylor expanded in x around inf 4.9%

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

13. Final simplification 4.9%

    \[\leadsto 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))