Hyperbolic tangent

Percentage Accurate: 9.3% → 99.5%

Time: 8.8s

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: 9.3% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{e^{x\_m} - t\_0}{e^{x\_m} + t\_0} \leq 0.2:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.13333333333333333 + {x\_m}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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)) 0.2)
      (*
       x_m
       (+
        1.0
        (*
         (pow x_m 2.0)
         (-
          (*
           (pow x_m 2.0)
           (+ 0.13333333333333333 (* (pow x_m 2.0) -0.05396825396825397)))
          0.3333333333333333))))
      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)) <= 0.2) {
		tmp = x_m * (1.0 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.13333333333333333 + (pow(x_m, 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	} 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)) <= 0.2d0) then
        tmp = x_m * (1.0d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * (0.13333333333333333d0 + ((x_m ** 2.0d0) * (-0.05396825396825397d0)))) - 0.3333333333333333d0)))
    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)) <= 0.2) {
		tmp = x_m * (1.0 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.13333333333333333 + (Math.pow(x_m, 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	} 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)) <= 0.2:
		tmp = x_m * (1.0 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * (0.13333333333333333 + (math.pow(x_m, 2.0) * -0.05396825396825397))) - 0.3333333333333333)))
	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)) <= 0.2)
		tmp = Float64(x_m * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.13333333333333333 + Float64((x_m ^ 2.0) * -0.05396825396825397))) - 0.3333333333333333))));
	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)) <= 0.2)
		tmp = x_m * (1.0 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * (0.13333333333333333 + ((x_m ^ 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	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], 0.2], N[(x$95$m * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.13333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $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)))) < 0.20000000000000001

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]

   if 0.20000000000000001 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 1.0%

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

      1. mul-1-neg 1.0%

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

      2. unsub-neg 1.0%

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

   5. Simplified 1.0%

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

   6. Taylor expanded in x around inf 1.0%

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

      1. associate--l+ 1.0%

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

      2. div-sub 1.0%

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

      3. distribute-rgt-in 1.0%

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

      4. *-lft-identity 1.0%

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

      5. cancel-sign-sub 1.0%

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

      6. mul-1-neg 1.0%

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

      7. *-commutative 1.0%

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

      8. unsub-neg 1.0%

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

      9. *-commutative 1.0%

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

      10. mul-1-neg 1.0%

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

      11. distribute-neg-frac2 1.0%

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

      12. associate-*l/ 1.0%

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

      13. associate-/l* 1.0%

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

      14. distribute-frac-neg2 1.0%

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

      15. *-rgt-identity 1.0%

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

      16. associate-*r/ 1.0%

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

      17. rgt-mult-inverse 1.0%

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

      18. metadata-eval 1.0%

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

      19. *-commutative 1.0%

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

   8. Simplified 1.0%

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

   10. Applied egg-rr 67.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 0.2:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.25)
    (*
     x_m
     (+
      1.0
      (*
       (pow x_m 2.0)
       (- (* (pow x_m 2.0) 0.13333333333333333) 0.3333333333333333))))
    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 * (1.0 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	} 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 * (1.0d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.13333333333333333d0) - 0.3333333333333333d0)))
    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 * (1.0 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	} 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 * (1.0 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333)))
	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(1.0 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.13333333333333333) - 0.3333333333333333))));
	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 * (1.0 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	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[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]

   if 1.25 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. div-sub 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. *-lft-identity 0.0%

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

      5. cancel-sign-sub 0.0%

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

      6. mul-1-neg 0.0%

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

      7. *-commutative 0.0%

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

      8. unsub-neg 0.0%

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

      9. *-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. distribute-neg-frac2 0.0%

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

      12. associate-*l/ 0.0%

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

      13. associate-/l* 0.0%

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

      14. distribute-frac-neg2 0.0%

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

      15. *-rgt-identity 0.0%

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

      16. associate-*r/ 0.0%

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

      17. rgt-mult-inverse 0.0%

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

      18. metadata-eval 0.0%

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

      19. *-commutative 0.0%

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

   8. Simplified 0.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.15)
    (* x_m (+ 1.0 (* (pow x_m 2.0) -0.3333333333333333)))
    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 * (1.0 + (pow(x_m, 2.0) * -0.3333333333333333));
	} 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 * (1.0d0 + ((x_m ** 2.0d0) * (-0.3333333333333333d0)))
    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 * (1.0 + (Math.pow(x_m, 2.0) * -0.3333333333333333));
	} 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 * (1.0 + (math.pow(x_m, 2.0) * -0.3333333333333333))
	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(1.0 + Float64((x_m ^ 2.0) * -0.3333333333333333)));
	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 * (1.0 + ((x_m ^ 2.0) * -0.3333333333333333));
	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[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)} \]

   if 1.1499999999999999 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. div-sub 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. *-lft-identity 0.0%

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

      5. cancel-sign-sub 0.0%

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

      6. mul-1-neg 0.0%

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

      7. *-commutative 0.0%

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

      8. unsub-neg 0.0%

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

      9. *-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. distribute-neg-frac2 0.0%

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

      12. associate-*l/ 0.0%

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

      13. associate-/l* 0.0%

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

      14. distribute-frac-neg2 0.0%

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

      15. *-rgt-identity 0.0%

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

      16. associate-*r/ 0.0%

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

      17. rgt-mult-inverse 0.0%

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

      18. metadata-eval 0.0%

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

      19. *-commutative 0.0%

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

   8. Simplified 0.0%

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

   10. Applied egg-rr 100.0%

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

Alternative 4: 99.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m + -0.33096296296296296 \cdot {x\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.15) (+ x_m (* -0.33096296296296296 (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 + (-0.33096296296296296 * 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 + ((-0.33096296296296296d0) * (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 + (-0.33096296296296296 * 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 + (-0.33096296296296296 * 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(-0.33096296296296296 * (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 + (-0.33096296296296296 * (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[(-0.33096296296296296 * 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 8.0%

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

   3. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{0.0023703703703703703} - 0.3333333333333333\right)\right) \]

   6. Taylor expanded in x around 0 98.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{-0.33096296296296296 \cdot {x}^{2}}\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-in 98.6%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.33096296296296296 \cdot {x}^{2}\right) \cdot x} \]

      2. *-un-lft-identity 98.6%

         \[\leadsto \color{blue}{x} + \left(-0.33096296296296296 \cdot {x}^{2}\right) \cdot x \]

      3. +-commutative 98.6%

         \[\leadsto \color{blue}{\left(-0.33096296296296296 \cdot {x}^{2}\right) \cdot x + x} \]

      4. associate-*l* 98.6%

         \[\leadsto \color{blue}{-0.33096296296296296 \cdot \left({x}^{2} \cdot x\right)} + x \]

      5. unpow2 98.6%

         \[\leadsto -0.33096296296296296 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \]

      6. pow3 98.6%

         \[\leadsto -0.33096296296296296 \cdot \color{blue}{{x}^{3}} + x \]

   8. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{-0.33096296296296296 \cdot {x}^{3} + x} \]

   if 1.1499999999999999 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. div-sub 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. *-lft-identity 0.0%

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

      5. cancel-sign-sub 0.0%

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

      6. mul-1-neg 0.0%

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

      7. *-commutative 0.0%

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

      8. unsub-neg 0.0%

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

      9. *-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. distribute-neg-frac2 0.0%

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

      12. associate-*l/ 0.0%

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

      13. associate-/l* 0.0%

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

      14. distribute-frac-neg2 0.0%

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

      15. *-rgt-identity 0.0%

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

      16. associate-*r/ 0.0%

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

      17. rgt-mult-inverse 0.0%

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

      18. metadata-eval 0.0%

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

      19. *-commutative 0.0%

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

   8. Simplified 0.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x + -0.33096296296296296 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 29.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m \cdot \left(1 + -0.33096296296296296 \cdot \left(x\_m \cdot x\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.15) (* x_m (+ 1.0 (* -0.33096296296296296 (* x_m 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 <= 1.15) {
		tmp = x_m * (1.0 + (-0.33096296296296296 * (x_m * 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 <= 1.15d0) then
        tmp = x_m * (1.0d0 + ((-0.33096296296296296d0) * (x_m * 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 <= 1.15) {
		tmp = x_m * (1.0 + (-0.33096296296296296 * (x_m * 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 <= 1.15:
		tmp = x_m * (1.0 + (-0.33096296296296296 * (x_m * 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 <= 1.15)
		tmp = Float64(x_m * Float64(1.0 + Float64(-0.33096296296296296 * Float64(x_m * 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 <= 1.15)
		tmp = x_m * (1.0 + (-0.33096296296296296 * (x_m * 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, 1.15], N[(x$95$m * N[(1.0 + N[(-0.33096296296296296 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{0.0023703703703703703} - 0.3333333333333333\right)\right) \]

   6. Taylor expanded in x around 0 98.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{-0.33096296296296296 \cdot {x}^{2}}\right) \]
   7. Step-by-step derivation

      1. unpow2 98.6%

         \[\leadsto x \cdot \left(1 + -0.33096296296296296 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Applied egg-rr 98.6%

      \[\leadsto x \cdot \left(1 + -0.33096296296296296 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   if 1.1499999999999999 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. div-sub 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. *-lft-identity 0.0%

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

      5. cancel-sign-sub 0.0%

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

      6. mul-1-neg 0.0%

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

      7. *-commutative 0.0%

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

      8. unsub-neg 0.0%

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

      9. *-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. distribute-neg-frac2 0.0%

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

      12. associate-*l/ 0.0%

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

      13. associate-/l* 0.0%

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

      14. distribute-frac-neg2 0.0%

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

      15. *-rgt-identity 0.0%

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

      16. associate-*r/ 0.0%

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

      17. rgt-mult-inverse 0.0%

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

      18. metadata-eval 0.0%

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

      19. *-commutative 0.0%

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

   8. Simplified 0.0%

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

   10. Applied egg-rr 100.0%

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

Alternative 6: 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 #s(literal 1 binary64) 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 8.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. div-sub 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. *-lft-identity 0.0%

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

      5. cancel-sign-sub 0.0%

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

      6. mul-1-neg 0.0%

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

      7. *-commutative 0.0%

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

      8. unsub-neg 0.0%

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

      9. *-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. distribute-neg-frac2 0.0%

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

      12. associate-*l/ 0.0%

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

      13. associate-/l* 0.0%

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

      14. distribute-frac-neg2 0.0%

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

      15. *-rgt-identity 0.0%

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

      16. associate-*r/ 0.0%

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

      17. rgt-mult-inverse 0.0%

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

      18. metadata-eval 0.0%

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

      19. *-commutative 0.0%

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

   8. Simplified 0.0%

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

   10. Applied egg-rr 100.0%

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

Alternative 7: 8.1% 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 #s(literal 1 binary64) 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 7.8%

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

3. Taylor expanded in x around 0 7.0%

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

   1. mul-1-neg 7.0%

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

   2. unsub-neg 7.0%

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

5. Simplified 7.0%

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

6. Taylor expanded in x around inf 8.3%

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

   1. associate--l+ 20.5%

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

   2. div-sub 19.3%

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

   3. distribute-rgt-in 19.3%

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

   4. *-lft-identity 19.3%

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

   5. cancel-sign-sub 19.3%

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

   6. mul-1-neg 19.3%

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

   7. *-commutative 19.3%

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

   8. unsub-neg 19.3%

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

   9. *-commutative 19.3%

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

   10. mul-1-neg 19.3%

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

   11. distribute-neg-frac2 19.3%

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

   12. associate-*l/ 19.3%

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

   13. associate-/l* 19.3%

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

   14. distribute-frac-neg2 19.3%

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

   15. *-rgt-identity 19.3%

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

   16. associate-*r/ 19.3%

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

   17. rgt-mult-inverse 19.3%

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

   18. metadata-eval 19.3%

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

   19. *-commutative 19.3%

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

8. Simplified 96.1%

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

10. Applied egg-rr 5.1%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))